#7623717

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×#509
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 ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1536 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530913);