#72322240

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623097	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,9,5,8,6,13)(3,7,12,15,11,10,14,16)(17,18,21,28,23,27,24,40)(19,30,32,46,34,26,35,42)(20,29,22,39,37,44,38,25)(31,43,48,45,36,47,33,41);(1,2,6,13,5,8,4,9)(3,10,14,15,11,7,12,16)(18,25,27,39)(20,22)(21,24)(26,41,30,45)(28,44,40,29)(31,48)(32,35)(33,36)(37,38)(42,47,46,43);(17,19)(18,26)(20,36)(21,32)(22,33)(23,34)(24,35)(25,41)(27,30)(28,42)(29,43)(31,37)(38,48)(39,45)(40,46)(44,47);(1,3)(2,7)(4,12)(5,11)(6,14)(8,10)(9,15)(13,16)(18,27)(25,39)(26,30)(28,40)(29,44)(41,45)(42,46)(43,47);(2,8)(3,11)(9,13)(12,14)(17,20)(19,31)(21,38)(22,24)(23,37)(25,29)(26,30)(28,40)(32,33)(34,36)(35,48)(39,44)(41,47)(43,45)

Polycyclic presentation:

2^10|a^2=f,b^2=fhij,f^2=h,g^2=i,i^2=j,b^a=bg,c^a=cj,d^a=dj,d^b=dhj,e^a=eg,e^b=ehi,e^c=ej,e^d=ehj,f^b=fj,f^e=fj,g^a=gi,g^b=gi,g^e=gij,i^a=ij,i^b=ij,i^e=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322260, #72322319
maximal normal subgroup of	#72322260, #72322319

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