#72228246

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#5020751
commutator, Frattini, cosocle	C₂⁶	C₂⁵
prime radical	#72211144	C₂
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

15372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^11|d^2=ghijk,c^a=ch,c^b=cf,d^a=di,d^b=dg,d^c=dfi,e^c=eghijk,e^d=efghjk,f^a=fj,f^d=fjk,f^e=fk,g^a=gjk,g^d=gk,g^e=gj,h^b=hj,h^d=hj,h^e=hjk,i^b=ijk,i^d=ijk,i^e=ik

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	UT(4,𝔽₄)
maximal normal subgroup of	UT(4,𝔽₄)

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);