#7623277

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622979
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 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1422 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530473);