#512183

order	512
nilpotency class	4
isoclinism	isoclinic class of #511887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

606 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,19,23,38)(12,24,27,32)(13,28,29,18)(14,30,22,16)(17,33,36,21)(25,37,26,40)(31,34,39,35);(2,5)(6,8)(9,11,13,22,15,23,29,14)(10,16,18,19,20,30,28,38)(12,21,25,39,27,33,26,31)(17,32,34,37,36,24,35,40);(1,3,4,7)(2,6,5,8)(9,11,13,22,15,23,29,14)(10,16,18,19,20,30,28,38)(12,21,25,39,27,33,26,31)(17,32,34,37,36,24,35,40);(9,12)(10,17)(11,21)(13,25)(14,31)(15,27)(16,32)(18,34)(19,37)(20,36)(22,39)(23,33)(24,30)(26,29)(28,35)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419379);