#512456

order	512
nilpotency class	4
isoclinism	isoclinic class of #511948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

674 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 218 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:

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

Polycyclic presentation:

2^9|a^2=i,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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419652);