#499638

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle, prime radical	C₂³	C₂²×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

734 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,8,9)(7,10,11,12)(13,14,17,22,33,40,44,43,19,24,35,42,18,23,34,41)(15,25,26,37,30,39,32,21,16,28,29,36,27,38,31,20);(1,2)(3,4)(6,9)(10,12)(13,15,19,16)(14,20,24,21)(17,31,35,32)(18,30,33,27)(22,38,42,39)(23,37,40,36)(25,43,28,41)(26,44,29,34);(2,4)(13,16,19,15)(14,21,24,20)(17,32,35,31)(18,27,33,30)(22,39,42,38)(23,36,40,37)(25,41,28,43)(26,34,29,44);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406834);