#4308512

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 12582912
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1698 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 1698 conjugacy classes of subgroups — too many to draw. Only its 610 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215708);