#508616

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	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

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

normal subgroups only

This group has 461 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fh,d^2=ei,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, 415812);