#512464

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=fhi,d^2=e,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=ghi,h^a=hi,d^b=di

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, 419660);