#327784

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1082 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1082 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,28,30)(27,33,34,40)(29,37,36,32)(31,38,39,35);(1,3,4,7)(2,5,6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(26,30)(31,39)(32,37)(33,40);(5,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,27,28,34)(26,31,30,39)(29,38,36,35)(32,40,37,33);(12,20)(15,22)(18,24)(21,23)(27,34)(31,39)(33,40)(35,38)

Polycyclic presentation:

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

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, 234980);