#72260809

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

49011 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|b^2=h,g^2=j,b^a=bh,d^c=di,e^c=ehi,f^d=fh,f^e=fj,g^a=gi,g^b=gij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72260932
maximal subgroup of	#72260932
maximal normal subgroup of	#72260932
maximal quotient of	#72260850

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