#433294

order	512
nilpotency class	3
isoclinism	isoclinic class of #422540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1051 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340490);