#509129

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	C₂²×#135	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

805 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)(9,10,13,19,29,38,39,36,15,21,30,35,14,20,28,18)(11,22,24,27,31,40,33,12,17,26,34,32,25,37,23,16);(5,6)(7,8)(9,11,15,17)(10,16,21,12)(13,23,30,33)(14,31,29,25)(18,22,36,26)(19,37,35,40)(20,27,38,32)(24,39,34,28);(2,4)(6,8)(9,12,15,16)(10,17,21,11)(13,26,30,22)(14,27,29,32)(18,33,36,23)(19,34,35,24)(20,31,38,25)(28,40,39,37);(2,4)(10,18)(11,23)(13,28)(14,29)(17,33)(19,20)(21,36)(22,37)(24,25)(26,40)(27,32)(30,39)(31,34)(35,38)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^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, 416325);