#512351

order	512
nilpotency class	4
isoclinism	isoclinic class of #512072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 494 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=eh,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,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, 419547);