#498047

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

506 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)(4,8)(5,7)(9,10,16,23,17,19,14,21,15,22,31,38,12,13,20,32)(11,24,29,18,25,37,33,30,28,26,39,36,34,40,27,35);(1,3,5,4)(2,6,7,8)(9,11,15,28)(10,18,22,36)(12,25,17,34)(13,30,19,35)(14,29,20,39)(16,33,31,27)(21,37,32,40)(23,26,38,24);(1,4,5,3)(2,6,7,8)(9,12,15,17)(10,19,22,13)(11,25,28,34)(14,16,20,31)(18,35,36,30)(21,38,32,23)(24,40,26,37)(27,29,33,39);(9,13,31,21,17,10,20,38,15,19,16,32,12,22,14,23)(11,26,29,36,25,40,33,35,28,24,39,18,34,37,27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405243);