#326708

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2656	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1636 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1636 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233904);