#511326

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,14)(13,16)(18,22)(23,30)(24,31)(25,29);(1,3,5,7)(2,6,8,4)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,29)(24,28)(25,27)(26,30)(31,32);(1,4,5,6)(2,7,8,3)(10,14)(13,15)(18,23)(20,28)(21,27)(22,30)(24,25)(26,32)(29,31);(2,8)(3,7)(11,16)(13,15)(18,22)(23,30)(24,31)(25,29)

Polycyclic presentation:

2^9|a^2=efi,b^2=efi,c^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

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, 418522);