#512417

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	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

560 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gh,h^a=hi,c^b=ci,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, 419613);