#511656

order	512
nilpotency class	4
isoclinism	isoclinic class of #511571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

494 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,12,15)(13,16,21,24)(19,23,20,22);(1,3)(2,5)(4,7)(6,8)(9,11,14,12)(10,15,17,18)(13,20,21,19)(16,23,24,22)(25,26,29,33)(27,30,36,39)(28,37,35,32)(31,40,38,34);(2,6)(5,8)(9,12,14,11)(10,15,17,18)(13,19,21,20)(16,23,24,22)(26,30)(27,34)(28,35)(31,32)(33,39)(36,40)(37,38);(3,7)(5,8)(9,13,14,21)(10,16,17,24)(11,19,12,20)(15,22,18,23)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=e,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418852);