#72241237

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

66925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=j,b^2=i,g^2=j,b^a=bh,d^a=di,d^b=di,f^a=fij,f^b=fij,g^a=gh,g^b=gij,g^c=gj,g^d=gi,g^e=gh,g^f=gij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241455
maximal normal subgroup of	#72241455

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