#7625131

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
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

1237 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,24)(20,33)(21,26)(22,40)(23,29)(25,37)(27,34)(28,39)(30,42)(31,36)(32,43)(35,47)(38,46)(41,44)(45,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,27)(21,25)(22,31)(23,32)(26,37)(28,42)(29,43)(30,39)(33,34)(35,44)(36,40)(38,45)(41,47)(46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,34,24,21,36,41,42,23,38,48,43,39,35,22,37)(18,25,40,47,28,32,45,46,29,30,44,31,26,19,27,33);(3,12)(7,10)(11,13)(15,16)(17,21,23,39)(18,26,29,28)(19,25,32,30)(20,35,38,36)(22,34,41,48)(24,37,43,42)(27,44,45,40)(31,33,47,46);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(17,21,23,39)(18,26,29,28)(19,30,32,25)(20,36,38,35)(22,34,41,48)(24,42,43,37)(27,44,45,40)(31,46,47,33)

Polycyclic presentation:

2^9|c^2=fgh,d^2=i,e^2=fi,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,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532327);