#406405

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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1309 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)(11,23)(12,28)(13,30)(14,21)(15,20)(16,22)(17,26)(18,29)(19,31)(24,36)(25,40)(27,37)(32,35)(33,39)(34,38);(2,5)(6,8)(9,11,14,26,16,27,15,32)(10,17,20,23,22,35,21,37)(12,24,19,34,30,39,29,40)(13,25,31,33,28,38,18,36);(9,12,15,29,16,30,14,19)(10,18,21,28,22,31,20,13)(11,24,26,34,27,39,32,40)(17,33,23,38,35,36,37,25);(1,3,4,7)(2,6,5,8)(9,13,14,31,16,28,15,18)(10,19,20,30,22,29,21,12)(11,25,26,33,27,38,32,36)(17,34,23,39,35,40,37,24)

Polycyclic presentation:

2^9|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, 313601);