#433146

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

979 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)(3,5)(4,6)(7,8)(9,10,15,21)(11,22)(12,27)(13,32,28,40)(14,19,34,20)(16,25)(17,29)(18,23,39,33)(24,36)(26,35)(30,38)(31,37);(1,3,4,7)(2,5,6,8)(9,11,14,25,15,26,34,24)(10,16,19,35,21,36,20,22)(12,28,30,33,31,13,29,23)(17,32,27,39,38,40,37,18);(2,6)(5,8)(9,12,15,31)(10,17,21,38)(11,23,26,33)(13,24,28,25)(14,29,34,30)(16,18,36,39)(19,37,20,27)(22,32,35,40);(9,13,14,23,15,28,34,33)(10,18,19,32,21,39,20,40)(11,12,24,29,26,31,25,30)(16,27,22,17,36,37,35,38)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=i,d^2=e,e^2=i,f^2=i,g^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,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, 340342);