#508579

order	512
nilpotency class	4
isoclinism	isoclinic class of #508579
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 ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 142 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)(9,10,12,18,15,20,14,17)(11,16,22,30,24,31,28,32)(13,19,25,34,27,36,26,33)(21,35,37,39,38,40,23,29);(2,5)(6,8)(9,11)(10,16)(12,22)(13,21)(14,28)(15,24)(17,32)(18,30)(19,35)(20,31)(23,26)(25,37)(27,38)(29,33)(34,39)(36,40);(1,3)(2,6)(4,7)(5,8)(10,17)(11,21,24,38)(12,14)(13,25)(16,29,31,39)(18,20)(22,23,28,37)(26,27)(30,40,32,35)(33,34);(3,7)(6,8)

Polycyclic presentation:

2^9|a^2=f,c^2=fh,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=ghi,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, 415775);