#511518

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 513 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ei,c^2=ei,d^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, 418714);