#7624017

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
prime radical	#61435	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1285 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gh,h^a=hi,c^b=ci,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, 7531213);