#406406

order	512
nilpotency class	3
isoclinism	isoclinic class of #398436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313602);