#72263173

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 23903 conjugacy classes of subgroups; only the 29475 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 24 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24);(17,19,21,20)(18,22,24,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14);(3,7)(4,8)(9,10)(11,12)(13,14)(15,16)(17,19,21,20)(18,23,24,22);(2,6)(4,8)(10,14)(12,15)(17,18)(19,22)(20,23)(21,24);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24);(1,4)(2,3)(5,8)(6,7)(9,12,13,15)(10,16,14,11)(17,20,21,19)(18,23,24,22)

Polycyclic presentation:

2^10|b^2=h,d^2=h,g^2=i,b^a=bh,e^a=ej,e^c=ej,e^d=ei,f^b=fh,f^d=fj,g^a=gh,g^d=gj,g^e=gj,g^f=gi

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72263231

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