#512461

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

Polycyclic presentation:

2^9|b^2=g,c^2=fh,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, 419657);