#509131

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2184	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

892 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)(9,10,15,29,27,36,26,17,14,20,12,18,13,19,31,34)(11,21,38,32,28,40,22,16,25,39,23,37,24,35,30,33);(5,6)(7,8)(9,11)(10,16)(12,22)(13,28)(14,25)(15,30)(17,21)(18,35)(19,37)(20,33)(23,26)(24,27)(29,40)(31,38)(32,36)(34,39);(2,4)(6,8);(2,4)(10,17)(11,22)(12,26)(13,27)(15,31)(18,19)(20,34)(21,35)(23,24)(25,30)(28,38)(29,36)(32,37)(39,40)

Polycyclic presentation:

2^9|a^2=ei,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=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, 416327);