#72309306

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 8529 conjugacy classes of subgroups; only the 29737 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:

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

Polycyclic presentation:

2^10|c^2=h,e^2=h,g^2=j,e^a=eh,e^b=ei,e^c=eh,e^d=ehj,f^a=fij,g^a=gj,g^e=gij

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309309
maximal normal subgroup of	#72309309

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