#513085

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
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
prime radical	#33733	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

347 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 122 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)(3,6)(4,5)(7,8)(10,15)(11,20)(12,21)(13,24)(16,26)(17,23)(18,34)(19,35)(22,33)(25,38)(27,28)(29,31)(30,40)(32,39)(36,37);(2,5)(6,8)(10,16)(15,26)(18,30)(19,31)(27,39)(28,32)(29,35)(34,40);(9,10,17,32,12,18,22,29,24,31,37,34,38,39,11,15,14,16,20,28,25,30,36,35,13,19,33,40,21,27,23,26);(1,3,4,7)(2,6,5,8)(10,16)(15,26)(18,30)(19,31)(27,39)(28,32)(29,35)(34,40)

Polycyclic presentation:

2^9|c^2=fgi,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,c^b=ci,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, 420281);