#513298

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×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 351 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,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^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, 420494);