#513215

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420411);