#72232618

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 8748 conjugacy classes of subgroups; only the 3101 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=h,d^2=ij,e^2=g,c^a=ch,c^b=chi,d^a=dgij,e^d=egij,f^b=fg,f^c=fhj,f^d=fj,f^e=fhj

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