#512457

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	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

684 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 194 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)(15,27)(16,21)(17,35)(18,38)(19,39)(20,26)(22,31)(23,28)(24,37)(25,36)(29,41)(30,40)(32,44)(33,43)(34,42);(6,9)(10,12)(13,15,28,33,19,30,37,34,20,31,18,29,36,32,17,16)(14,21,35,44,25,41,38,22,26,42,24,40,39,43,23,27);(1,2,3,4)(13,15,28,33,19,30,37,34,20,31,18,29,36,32,17,16)(14,21,35,44,25,41,38,22,26,42,24,40,39,43,23,27);(5,7,8,11)(6,10,9,12)(13,16,17,32,36,29,18,31,20,34,37,30,19,33,28,15)(14,22,23,41,39,44,24,21,26,27,38,43,25,40,35,42)

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

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, 419653);