#7625023

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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

1386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,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,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532219);