#72228243

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×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

15332 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)(4,9)(5,10)(6,11)(7,13)(12,28)(14,29)(15,30)(16,42)(17,31)(18,32)(19,48)(20,35)(21,36)(22,34)(23,37)(24,54)(25,41)(26,39)(27,40)(33,43)(38,46)(44,57)(45,56)(47,58)(49,61)(50,64)(51,63)(52,62)(53,60)(55,59);(1,3)(2,8)(4,19,22,14)(5,24,26,15)(6,18)(7,17)(9,33,34,56)(10,38,39,58)(11,32)(12,42)(13,31)(16,28)(20,50,52,44)(21,43,23,45)(25,46,27,47)(29,36,48,37)(30,41,54,40)(35,61,62,59)(49,53,55,51)(57,63,64,60);(9,34)(10,39)(14,43)(15,46)(19,45)(24,47)(29,56)(30,58)(33,48)(35,62)(36,37)(38,54)(40,41)(44,49)(50,55)(57,59)(60,63)(61,64);(1,4)(2,9)(3,14)(5,20)(6,21)(7,23)(8,29)(10,35)(11,36)(12,34)(13,37)(15,44)(16,19)(17,43)(18,45)(22,28)(24,55)(25,53)(26,51)(27,52)(30,57)(31,33)(32,56)(38,64)(39,63)(40,62)(41,60)(42,48)(46,50)(47,49)(54,59)(58,61);(1,5)(2,10)(3,15)(4,20)(6,27)(7,26)(8,30)(9,35)(11,40)(12,41)(13,39)(14,44)(16,47)(17,24)(18,46)(19,49)(21,52)(22,53)(23,51)(25,28)(29,57)(31,54)(32,38)(33,59)(34,60)(36,62)(37,63)(42,58)(43,55)(45,50)(48,61)(56,64)

Polycyclic presentation:

2^11|b^2=i,b^a=bh,c^a=ci,c^b=chi,d^b=df,d^c=dg,e^b=eg,e^c=efg,f^a=fj,f^b=fjk,g^a=gjk,g^b=gk,h^d=hj,h^e=hjk,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, 2, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*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);