#512470

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
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 ◁ C₄³ ◁ C₄²×C₈ ◁ 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)(13,14)(15,30)(16,22)(17,38)(18,27)(19,26)(20,40)(21,29)(23,32)(24,35)(25,31)(28,39)(33,42)(34,41)(36,43)(37,44);(6,9)(10,12)(13,15,31,37,20,34,18,32,21,35,19,33,39,36,17,16)(14,22,38,43,28,42,26,24,29,23,27,41,40,44,25,30);(1,2,3,4)(13,15,19,33,20,34,17,16,21,35,31,37,39,36,18,32)(14,23,27,43,28,44,25,24,29,22,38,41,40,42,26,30);(5,7,8,11)(6,10,9,12)(13,16,17,36,39,33,19,35,21,32,18,34,20,37,31,15)(14,24,25,42,40,43,27,22,29,30,26,44,28,41,38,23)

Polycyclic presentation:

2^9|b^2=gi,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,c^b=ci,d^c=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, 419666);