#72256162

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 20579 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) = 128

A minimal faithful permutation representation — 7 generators on 24 points:

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

Polycyclic presentation:

2^10|b^2=i,g^2=j,b^a=bi,d^b=dh,d^c=di,e^a=eh,e^b=eh,e^c=ej,f^a=fi,f^b=fi,g^a=gi,g^d=gh,g^f=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256171, #72263046, #72265968
maximal normal subgroup of	#72256171, #72263046, #72265968
maximal quotient of	#72266267, #72266585, #72266587

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