#496048

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
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	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

570 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)(9,10,15,22,17,23,16,32)(11,19,26,34,28,38,27,33)(12,18,29,35,24,36,25,37)(13,20,30,39,31,40,14,21);(2,5)(6,8)(9,11,14,25,16,27,31,24,17,28,30,29,15,26,13,12)(10,18,20,34,22,35,39,38,23,36,40,33,32,37,21,19);(1,3,4,7)(2,6,5,8)(9,12,13,26,15,29,30,28,17,24,31,27,16,25,14,11)(10,19,21,37,32,33,40,36,23,38,39,35,22,34,20,18);(9,13,15,30,17,31,16,14)(10,20,22,39,23,40,32,21)(11,24,26,25,28,12,27,29)(18,33,35,19,36,34,37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403244);