#72262907

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 39796 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=i,c^2=j,d^2=j,f^2=j,c^a=cij,d^a=dij,e^a=eh,e^b=eh,e^d=ehij,f^a=fij,f^b=fh,f^d=fj,f^e=fh,g^a=gj,g^b=gi,g^c=gj,g^d=gj,g^f=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263046
maximal normal subgroup of	#72263046

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