#508582

order	512
nilpotency class	4
isoclinism	isoclinic class of #508581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

620 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415778);