#509693

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33607	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416889);