#329588

order	512
nilpotency class	3
isoclinism	isoclinic class of #329587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 836 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)(3,6)(4,5)(7,8)(9,10,12,15,13,16,21,24)(11,17,18,22,19,23,20,14)(25,26)(27,30)(28,32)(29,31)(33,37)(34,39)(35,38)(36,40);(2,5)(6,8)(25,27)(26,30)(28,33)(29,35)(31,38)(32,37)(34,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11,13,19)(10,14,16,22)(12,20,21,18)(15,23,24,17)(27,33)(28,36)(30,37)(32,40)(34,35)(38,39);(3,7)(6,8)(26,31)(30,38)(32,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236784);