#509138

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	#25777	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

834 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)(9,10,17,23,12,21,31,33,15,11,14,20,16,18,25,29)(13,26,39,35,32,38,30,22,28,36,27,37,24,40,34,19);(5,6)(7,8)(9,11,15,10)(12,21,16,18)(13,26,28,36)(14,29,17,33)(19,27,22,39)(20,31,23,25)(24,38,32,40)(30,37,34,35);(2,4)(6,8)(9,12,15,16)(10,18,11,21)(13,24,28,32)(14,25,17,31)(19,35,22,37)(20,33,23,29)(26,38,36,40)(27,30,39,34);(2,4)(9,13,15,28)(10,19,11,22)(12,24,16,32)(14,30,17,34)(18,35,21,37)(20,38,23,40)(25,39,31,27)(26,33,36,29)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^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, 416334);