#509589

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13304	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

716 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|d^2=ehi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,h^a=hi,c^b=cg,d^b=de,e^b=eh,h^b=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, 416785);