#328915

order	512
nilpotency class	3
isoclinism	isoclinic class of #326835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 682 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:

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

Polycyclic presentation:

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

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, 236111);