#393927

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3881 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)(3,4)(5,6)(7,8)(9,10)(11,21)(12,15)(13,29)(14,20)(16,19)(17,24)(18,32)(22,25)(23,37)(26,27)(28,30)(31,34)(33,39)(35,40)(36,38);(2,4)(9,11,15,25)(10,17,12,27)(13,23,31,39)(14,26,16,24)(18,33,28,37)(19,22,20,21)(29,36,34,35)(30,40,32,38);(6,8)(9,12,15,10)(11,22,25,21)(13,28,31,18)(14,20,16,19)(17,26,27,24)(23,36,39,35)(29,32,34,30)(33,38,37,40);(9,13)(10,18)(11,23)(12,28)(14,32)(15,31)(16,30)(17,33)(19,29)(20,34)(21,35)(22,36)(24,40)(25,39)(26,38)(27,37)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301123);