#508380

order	512
nilpotency class	4
isoclinism	isoclinic class of #508286
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

677 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 677 conjugacy classes of subgroups — too many to draw. Only its 146 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)(9,10,13,18,15,19,29,14)(11,16,21,32,23,33,22,30)(12,24,26,34,27,35,28,17)(20,25,36,39,38,40,37,31);(2,5)(6,8)(9,11)(10,16)(12,25)(13,21)(14,30)(15,23)(17,20)(18,32)(19,33)(22,29)(24,36)(26,39)(27,40)(28,31)(34,38)(35,37);(3,7)(6,8)(9,12,15,27)(10,17,19,34)(11,20,23,38)(13,28,29,26)(14,24,18,35)(16,31,33,39)(21,37,22,36)(25,32,40,30);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,f^c=fi,g^c=gh,h^c=hi

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, 415576);