#325810

order	512
nilpotency class	3
isoclinism	isoclinic class of #288715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

552 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233006);