#72265482

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 11493 conjugacy classes of subgroups; only the 29713 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)(6,8)(11,15)(14,16)(17,18,21,23)(19,20,22,24);(11,15)(14,16)(17,19)(18,20)(21,22)(23,24);(3,7)(6,8)(10,13)(14,16);(9,10)(11,14)(12,13)(15,16);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16);(17,20)(18,22)(19,23)(21,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)

Polycyclic presentation:

2^10|a^2=i,d^c=dh,e^a=ej,e^b=eh,f^b=fi,g^c=gj

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265557, #72265842
maximal normal subgroup of	#72265557, #72265842
maximal quotient of	#72266546

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