#290889

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

875 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 875 conjugacy classes of subgroups — too many to draw. Only its 136 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)(11,15)(12,16)(13,14)(17,18)(19,24)(20,21)(22,25)(23,26)(27,30)(28,32)(29,31);(1,3,4,8)(2,5,6,7)(11,16)(12,15)(17,19)(18,24)(20,30)(21,27)(22,28)(23,29)(25,32)(26,31);(9,11,15,10,13,16,12,14)(17,20,21,18,23,25,22,26)(19,24,27,32,29,31,28,30);(2,6)(5,7)(17,21,23,22)(18,20,26,25)(19,27,29,28)(24,30,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198085);