#291090

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

935 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 935 conjugacy classes of subgroups — too many to draw. Only its 156 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,16,14,12)(17,18,22,21)(19,23,26,29)(20,24,28,30)(25,31,32,27);(1,3,4,8)(2,5,6,7)(11,14)(12,16)(17,19,22,26)(18,23,21,29)(20,27,28,31)(24,25,30,32);(9,11,12,10,13,14,16,15)(17,20,22,28)(18,24,21,30)(19,25,26,32)(23,31,29,27);(2,6)(5,7)(17,18,22,21)(19,23,26,29)(20,24,28,30)(25,31,32,27)

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=fi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198286);