#502118

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409314);