#7625021

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

1398 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)(9,10,14,21)(11,23,26,17)(12,28,32,39)(13,19,16,20)(15,36,35,38)(18,37,25,33)(22,34,24,29)(27,31,40,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,24)(13,18)(14,26)(15,27)(16,25)(19,33)(20,37)(21,23)(22,32)(28,34)(29,39)(30,36)(31,38)(35,40);(9,12,29,23,16,31,35,37,14,32,34,17,13,30,15,33)(10,18,36,27,19,11,24,39,21,25,38,40,20,26,22,28);(2,5)(6,8)(9,13,14,16)(10,19,21,20)(11,25,26,18)(12,30,32,31)(15,34,35,29)(17,37,23,33)(22,36,24,38)(27,39,40,28);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,20,21,19)(11,25,26,18)(12,31,32,30)(15,34,35,29)(17,33,23,37)(22,38,24,36)(27,39,40,28)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^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,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, 7532217);