#7624792

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

1202 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)(19,28)(20,26)(21,34)(22,32)(23,25)(24,30)(27,36)(29,46)(31,45)(33,42)(35,40)(37,44)(38,43)(39,47)(41,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18)(19,24)(20,32)(21,27)(22,26)(23,25)(28,30)(29,41)(31,39)(33,44)(34,36)(35,43)(37,42)(38,40)(45,47)(46,48);(17,19,23,31)(18,24,25,39)(20,29,22,38)(21,35,37,48)(26,40,32,41)(27,43,42,46)(28,44,45,34)(30,33,47,36);(3,10)(6,9)(11,16)(14,15)(18,25)(24,39)(26,32)(27,42)(28,45)(34,44)(40,41)(43,46);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,20,23,22)(18,26,25,32)(19,29,31,38)(21,33,37,36)(24,40,39,41)(27,34,42,44)(28,46,45,43)(30,35,47,48)

Polycyclic presentation:

2^9|a^2=f,c^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, 7531988);