#512311

order	512
nilpotency class	4
isoclinism	isoclinic class of #511960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

577 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 577 conjugacy classes of subgroups — too many to draw. Only its 156 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)(10,15)(11,20)(12,16)(13,24)(17,32)(18,35)(19,29)(21,23)(22,30)(25,31)(26,33)(27,28)(34,38)(36,37);(1,3,4,7)(2,5,6,8)(9,10,16,28,13,18,25,29,14,19,31,35,24,27,12,15)(11,17,30,39,22,32,20,33,23,34,37,40,36,38,21,26);(2,6)(5,8);(9,11,12,21,24,36,31,37,14,23,25,20,13,22,16,30)(10,17,15,26,27,38,35,40,19,34,29,33,18,32,28,39)

Polycyclic presentation:

2^9|b^2=efh,d^2=e,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,h^a=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419507);