#72239743

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

46255 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=h,d^2=i,f^2=hj,g^2=j,b^a=bj,c^a=ch,c^b=chj,d^a=di,e^a=ehi,e^b=ei,f^a=fhj,f^b=fhj,g^a=ghj,g^b=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72239805
maximal subgroup of	#72239752, #72239805
maximal normal subgroup of	#72239752, #72239805

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