#509148

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

817 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)(9,10,15,29,27,34,25,16,13,17,12,19,14,20,30,32)(11,21,38,36,28,40,26,18,23,39,22,35,24,37,31,33);(5,6)(7,8)(10,16)(11,22)(12,25)(14,27)(15,30)(17,32)(18,35)(19,20)(23,38)(24,31)(26,28)(29,34)(33,36)(37,40);(2,4)(6,8)(10,17)(16,32)(18,33)(19,29)(20,34)(21,39)(35,36)(37,40);(2,4)(9,11,13,23)(10,18,17,33)(12,26,15,31)(14,28,27,24)(16,21,32,39)(19,37,29,40)(20,35,34,36)(22,30,38,25)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416344);