#72328083

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 39849 conjugacy classes of subgroups; only the 29385 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|f^2=hj,g^2=ij,b^a=bh,c^b=chi,f^a=fhij,f^c=fj,f^d=fhij,f^e=fhj,g^a=gj,g^c=gj,g^d=gi,g^e=gij,g^f=ghj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72328397, #72328873, #72329191

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