#72322398

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	#7624075	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17330 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2271 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322467, #72322909, #72343989
maximal normal subgroup of	#72322467, #72322909, #72343989

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