#72333792

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 20095 conjugacy classes of subgroups; only the 29337 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 7 generators on 32 points:

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

Polycyclic presentation:

2^10|b^a=bj,c^b=ch,f^b=fi,f^c=fi,f^d=fhj,f^e=fhij,g^a=gi,g^b=ghij,g^c=gi,g^d=gh,g^e=gij

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72333951, #72334051, #72334158

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