#509600

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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

710 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 144 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)(12,24)(13,15)(14,19)(16,21)(17,34)(18,23)(22,31)(25,37)(26,40)(27,30)(28,36)(29,32)(33,39)(35,38);(5,6)(7,8)(10,15)(11,21)(12,25)(13,23)(16,18)(19,32)(20,30)(22,38)(24,40)(26,39)(27,29)(28,34)(31,36)(33,35);(2,4)(6,8)(9,11,14,21)(10,16,19,20)(12,26,28,22)(13,23,29,27)(15,30,32,18)(17,35,37,33)(24,31,36,40)(25,38,34,39);(2,4)(9,12,27,38,11,22,13,25,14,28,23,39,21,26,29,34)(10,17,32,40,16,33,18,36,19,37,15,31,20,35,30,24)

Polycyclic presentation:

2^9|c^2=i,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,d^c=di

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, 416796);