#512443

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

776 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 234 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)(15,23)(16,21)(17,25)(18,22)(19,24)(20,35)(26,32)(27,39)(28,42)(29,44)(30,37)(31,36)(33,41)(34,40)(38,43);(6,9)(10,12)(13,15,19,28,20,29,33,31,18,27,34,43,32,30,17,16)(14,21,25,37,26,38,40,39,22,36,41,44,35,42,24,23);(1,2,3,4)(14,22)(21,36)(23,39)(24,40)(25,41)(26,35)(37,44)(38,42);(5,7,8,11)(6,10,9,12)(13,16,17,30,32,43,34,27,18,31,33,29,20,28,19,15)(14,23,24,42,35,44,41,36,22,39,40,38,26,37,25,21)

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

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, 419639);