#499065

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1212 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 250 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)(3,4)(6,7)(9,10,15,11)(12,22,24,23)(13,27,32,28)(14,21,30,18)(16,34,19,25)(17,36,20,31)(26,39,33,37)(29,38,40,35);(5,6)(7,8)(9,11,15,10)(12,23,24,22)(13,28,32,27)(14,18,30,21)(16,25,19,34)(17,31,20,36)(26,37,33,39)(29,35,40,38);(2,4)(5,7,8,6)(9,12)(10,16)(11,19)(13,29)(14,33)(15,24)(17,37)(18,38)(20,39)(21,35)(22,27)(23,28)(25,31)(26,30)(32,40)(34,36);(12,24)(16,19)(22,23)(25,34)(26,33)(29,40)(35,38)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406261);