#7625061

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
prime radical	#62495	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1161 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)(19,29)(20,33)(21,26)(22,40)(23,42)(24,31)(25,36)(27,41)(28,39)(30,34)(32,37)(35,46)(38,43)(44,45)(47,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,25)(26,29)(27,35)(28,36)(33,41)(34,46)(37,43)(38,47)(39,44)(40,48)(42,45);(17,20,34,24,23,37,22,36,21,35,41,29,39,48,38,45)(18,25,43,32,28,31,27,33,26,44,40,47,42,19,30,46);(3,10)(6,9)(11,16)(14,15)(19,31)(20,35)(24,29)(25,44)(32,47)(33,46)(36,45)(37,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,26)(19,31)(25,44)(27,30)(28,42)(32,47)(33,46)(40,43)

Polycyclic presentation:

2^9|a^2=fi,c^2=ghi,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, 7532257);