#7624801

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1230 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318797, #72320093, #72320482
maximal normal subgroup of	#72318797, #72320093, #72320482

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531997);