#72322339

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	#7624079	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17330 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2213 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=h,e^2=h,f^2=h,g^2=i,i^2=j,b^a=bgi,c^b=cj,d^a=dgj,d^b=dh,d^c=dhj,e^b=eh,e^c=ehj,f^a=fj,g^b=gi,g^d=gij,i^b=ij,i^d=ij

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322376
maximal normal subgroup of	#72322376

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