#509149

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×#2186	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

826 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)(9,10,16,32,30,39,29,19,14,21,13,20,15,22,34,37)(11,23,36,35,31,40,24,17,18,27,25,28,26,38,33,12);(5,6)(7,8)(9,11)(10,17)(12,21)(13,24)(14,18)(15,31)(16,33)(19,23)(20,38)(22,28)(25,29)(26,30)(27,37)(32,40)(34,36)(35,39);(2,4)(6,8)(9,12,14,17)(10,18,21,11)(13,27,16,23)(15,28,30,35)(19,33,37,24)(20,36,32,25)(22,31,39,26)(29,40,34,38);(2,4)(10,19)(11,24)(13,29)(15,30)(16,34)(18,33)(20,22)(21,37)(23,38)(25,26)(27,40)(28,35)(31,36)(32,39)

Polycyclic presentation:

2^9|a^2=ei,c^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, 416345);