#513147

order	512
nilpotency class	5
isoclinism	isoclinic class of #513086
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 ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 341 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:

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

Polycyclic presentation:

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

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, 420343);