#512296

order	512
nilpotency class	4
isoclinism	isoclinic class of #511890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

638 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^d=ghi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419492);