#512271

order	512
nilpotency class	4
isoclinism	isoclinic class of #512072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,d^2=eh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^b=gi,e^d=ei,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, 419467);