#72238579

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 23376 conjugacy classes of subgroups; only the 29425 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:

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

Polycyclic presentation:

2^10|b^2=h,d^2=h,e^2=i,f^2=hj,b^a=bh,c^b=ch,d^a=dh,d^c=dhj,e^c=ei,f^c=fhj,f^e=fhij,g^e=gj,g^f=gi

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260847
maximal normal subgroup of	#72260847

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, 5, F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*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);