#472521

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1094 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1094 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379717);