#72304484

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 14031 conjugacy classes of subgroups; only the 29707 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=j,c^2=j,d^2=h,f^2=h,g^2=i,c^b=ci,e^a=ej,e^b=ej,e^c=ej,e^d=ej,f^a=fhj,f^b=fhj,f^c=fhj,f^d=fhj,f^e=fh,g^a=gij,g^b=gij,g^c=gj,g^e=gi

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304523
maximal normal subgroup of	#72304523

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