#509899

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417095);