#7625003

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

1391 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)(9,10,13,18)(11,21,23,16)(12,26,28,37)(14,33,31,38)(15,34,32,20)(17,35,25,29)(19,36,22,27)(24,30,39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,32)(18,21)(19,28)(20,29)(26,36)(27,37)(30,38)(31,39)(33,40)(34,35);(9,12,27,21,15,30,31,35,13,28,36,16,32,40,14,29)(10,17,33,24,20,11,22,37,18,25,38,39,34,23,19,26);(2,5)(6,8)(11,23)(12,28)(16,21)(17,25)(24,39)(26,37)(29,35)(30,40);(1,3,4,7)(2,6,5,8)(10,18)(11,23)(17,25)(19,22)(20,34)(24,39)(26,37)(33,38)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,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, 7532199);