#513038

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

440 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 150 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)(10,11)(12,13)(14,24)(15,30)(17,21)(18,22)(19,36)(20,35)(23,34)(25,39)(26,29)(27,28)(31,38)(32,37)(33,40);(2,5)(6,8)(9,10,13,18,14,19,27,35,30,34,39,40,38,37,26,21,16,17,29,32,31,33,25,23,15,20,28,36,24,22,12,11);(1,3,4,7)(2,6,5,8)(9,11,12,22,24,36,28,20,15,23,25,33,31,32,29,17,16,21,26,37,38,40,39,34,30,35,27,19,14,18,13,10);(10,17)(11,21)(18,32)(19,33)(20,34)(22,37)(23,35)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420234);