#512254

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

492 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)(3,5)(4,6)(7,8)(9,10,17,22)(11,14,21,16)(12,23,27,20)(13,28,32,35)(15,33,26,18)(19,25,37,31)(24,34,36,40)(29,39,30,38);(1,3,4,7)(2,5,6,8)(9,11,14,22,17,21,16,10)(12,23,26,33,27,20,15,18)(13,24,29,37,32,36,30,19)(25,39,40,28,31,38,34,35);(2,6)(5,8)(9,12,14,26,17,27,16,15)(10,18,11,23,22,33,21,20)(13,25,29,40,32,31,30,34)(19,35,24,39,37,28,36,38);(9,13)(10,19)(11,24)(12,25)(14,29)(15,34)(16,30)(17,32)(18,35)(20,38)(21,36)(22,37)(23,39)(26,40)(27,31)(28,33)

Polycyclic presentation:

2^9|a^2=i,b^2=efh,c^2=e,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,g^c=gi,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, 419450);