#72342321

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 9105 conjugacy classes of subgroups; only the 29493 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:

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

Polycyclic presentation:

2^10|d^2=j,f^2=hj,g^2=j,d^a=dh,d^c=dj,e^b=ehi,e^d=ei,f^b=fhi,f^c=fhj,f^d=fi,g^a=gi,g^b=gj,g^d=gj,g^e=gij,g^f=gij

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342349
maximal normal subgroup of	#72342349

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