#135220

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1817
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=d,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^b=fi,e^c=ei,f^c=fg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42416);