#72232652

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 9096 conjugacy classes of subgroups; only the 3113 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=j,e^2=hj,c^a=ch,c^b=cg,d^c=dj,e^b=eghi,e^d=ej,f^a=fghi,f^d=fij,f^e=fghi

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232676
maximal normal subgroup of	#72232676

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