#72340312

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,11)(5,8)(7,13)(9,14)(10,15)(12,16)(22,28)(23,30)(26,31)(29,32);(1,2)(3,6)(4,7,13,11)(5,8)(9,14,12,16)(10,15)(17,18)(19,22,25,28)(20,23,27,30)(21,24)(26,32)(29,31);(1,3,5,10)(2,6,8,15)(4,12,13,9)(7,14,11,16)(19,25)(20,27);(1,4,5,13)(2,7)(3,9,10,12)(6,14)(8,11)(15,16)(17,19)(18,22,24,28)(20,26)(21,25)(23,29,30,32)(27,31);(4,13)(6,15)(7,11)(9,12)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)

Polycyclic presentation:

2^10|b^2=f,c^2=ij,d^2=gij,b^a=bf,c^b=cf,d^a=dfh,d^b=dfg,d^c=dgi,e^a=eh,e^b=ehj,e^c=egj,e^d=eij,f^d=fi,f^e=fj,g^a=gij,g^b=gi,h^b=hj,h^c=hij,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340328, #72340368, #72340495
maximal normal subgroup of	#72340328, #72340368, #72340495

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> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
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);