#509128

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

796 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)(9,10,13,18,30,36,40,39,15,20,32,37,14,19,31,22)(11,21,23,27,33,29,17,12,25,26,38,34,24,35,28,16);(5,6)(7,8)(9,11,15,25)(10,16,20,12)(13,28,32,17)(14,33,30,24)(18,35,37,29)(19,27,36,34)(21,39,26,22)(23,40,38,31);(2,4)(6,8);(2,4)(9,12,15,16)(10,17,20,28)(11,22,25,39)(13,29,32,35)(14,34,30,27)(18,33,37,24)(19,38,36,23)(21,31,26,40)

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=i,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, 416324);