#7624555

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624235
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 ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1357 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)(9,10,14,17)(11,15,20,30)(12,16,25,31)(13,26,24,19)(18,33,32,23)(21,27,34,39)(22,37,35,29)(28,40,38,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,27)(17,30)(18,28)(19,29)(23,36)(24,35)(25,34)(26,37)(31,39)(32,38)(33,40);(10,16)(12,23)(13,24)(15,27)(17,31)(18,19)(21,36)(22,35)(25,33)(26,32)(28,29)(30,39)(34,40)(37,38);(2,5)(6,8)(10,17)(11,20)(16,31)(18,32)(19,26)(21,34)(22,35)(36,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,g^c=gh,h^c=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313320, #72315864, #72315916
maximal normal subgroup of	#72313320, #72315864, #72315916

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531751);