#7624797

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

1203 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)(19,22)(20,31)(21,28)(23,37)(24,26)(25,34)(27,39)(29,45)(30,35)(32,47)(33,46)(36,42)(38,44)(40,43)(41,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,22,18)(20,29,33,25)(21,28,24,26)(23,30,36,27)(31,34,46,45)(32,44,40,41)(35,37,39,42)(38,47,48,43);(17,20)(18,25)(19,29)(21,32)(22,33)(23,38)(24,40)(26,41)(27,43)(28,44)(30,47)(31,37)(34,39)(35,45)(36,48)(42,46);(3,10)(6,9)(11,16)(14,15)(17,21,22,24)(18,26,19,28)(20,32,33,40)(23,35,36,39)(25,41,29,44)(27,42,30,37)(31,43,46,47)(34,38,45,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,19)(25,29)(26,28)(27,30)(31,46)(37,42)(41,44)(43,47)

Polycyclic presentation:

2^9|a^2=fi,b^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531993);