#512449

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	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 764 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) = 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)(17,34)(18,22)(19,38)(20,39)(23,28)(24,37)(25,36)(26,35)(29,42)(30,41)(31,44)(32,40)(33,43);(6,9)(10,12)(13,15,19,29,20,30,17,16,18,28,37,33,35,31,36,32)(14,21,25,41,26,42,24,27,22,40,34,44,39,43,38,23);(1,2,3,4)(14,22)(21,40)(23,27)(24,38)(25,34)(26,39)(41,44)(42,43);(5,7,8,11)(6,10,9,12)(13,16,17,31,35,29,19,28,18,32,36,30,20,33,37,15)(14,23,24,42,39,44,25,21,22,27,38,43,26,41,34,40)

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,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, 419645);