H(5,𝔽₄) #72228232

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

Heisenberg

H(5,𝔽₄)

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

430453 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 8 generators on 64 points:

(2,8)(3,14)(5,23)(9,10)(11,33)(12,28)(13,32)(15,18)(16,39)(17,45)(21,46)(24,25)(27,30)(29,35)(31,34)(36,55)(38,54)(40,42)(41,43)(44,61)(47,48)(49,52)(51,57)(59,60);(2,9)(4,19)(5,23)(8,10)(11,30)(12,28)(13,29)(16,41)(17,42)(20,22)(24,25)(27,33)(31,34)(32,35)(36,49)(37,50)(38,54)(39,43)(40,45)(51,57)(52,55)(53,56)(58,63)(62,64);(3,15)(4,19)(5,24)(11,33)(12,34)(13,32)(14,18)(17,45)(20,22)(21,48)(23,25)(27,30)(28,31)(29,35)(37,50)(38,51)(40,42)(44,59)(46,47)(53,56)(54,57)(58,63)(60,61)(62,64);(2,10)(3,15)(4,20)(5,23)(8,9)(12,28)(14,18)(16,43)(19,22)(21,48)(24,25)(31,34)(36,52)(37,56)(38,54)(39,41)(44,59)(46,47)(49,55)(50,53)(51,57)(58,62)(60,61)(63,64);(1,2)(3,11)(4,12)(5,13)(6,8)(7,10)(9,26)(14,30)(15,33)(16,36)(17,37)(18,27)(19,34)(20,31)(21,38)(22,28)(23,35)(24,32)(25,29)(39,52)(40,53)(41,55)(42,56)(43,49)(44,58)(45,50)(46,57)(47,54)(48,51)(59,63)(60,64)(61,62);(1,3)(2,11)(4,16)(5,17)(6,18)(7,15)(8,27)(9,30)(10,33)(12,36)(13,37)(14,26)(19,41)(20,39)(21,44)(22,43)(23,42)(24,40)(25,45)(28,49)(29,50)(31,52)(32,53)(34,55)(35,56)(38,58)(46,60)(47,59)(48,61)(51,62)(54,63)(57,64);(1,4)(2,12)(3,16)(5,21)(6,22)(7,19)(8,28)(9,31)(10,34)(11,36)(13,38)(14,39)(15,41)(17,44)(18,43)(20,26)(23,46)(24,47)(25,48)(27,49)(29,51)(30,52)(32,54)(33,55)(35,57)(37,58)(40,59)(42,60)(45,61)(50,62)(53,63)(56,64);(1,5)(2,13)(3,17)(4,21)(6,25)(7,23)(8,29)(9,32)(10,35)(11,37)(12,38)(14,40)(15,42)(16,44)(18,45)(19,46)(20,47)(22,48)(24,26)(27,50)(28,51)(30,53)(31,54)(33,56)(34,57)(36,58)(39,59)(41,60)(43,61)(49,62)(52,63)(55,64)

Polycyclic presentation:

2^10|e^a=ei,e^b=eij,e^d=ej,f^a=fij,f^c=fj,f^d=fj,g^b=gj,g^c=gj,g^d=gij,h^a=hj,h^b=hj,h^c=hij,h^d=hj

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72241426, #72241456, #72268535
Fitting subgroup of	#72235028, #72241426, #72241456, …
maximal subgroup of	#72228235, #72235015, #72235028, …
maximal normal subgroup of	#72228235, #72235015, #72235028, …
maximal quotient of	#72228234, #72306591

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

H(5,𝔽4) #72228232