#496050

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

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,c^b=ci,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, 403246);