#7624716

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1200 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)(19,27)(20,30)(21,32)(23,35)(25,40)(26,39)(28,45)(29,44)(31,34)(33,43)(36,47)(37,48)(38,46)(41,42);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,35)(28,37)(29,36)(30,40)(31,41)(32,39)(33,38)(34,42)(43,46)(44,47)(45,48);(17,19,22,29)(18,23,24,36)(20,31,33,45)(21,34,32,28)(25,41,38,48)(26,42,39,37)(27,43,44,30)(35,46,47,40);(3,10)(6,9)(11,16)(14,15);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,24)(23,36)(25,38)(26,39)(35,47)(37,42)(40,46)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531912);