#416792

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eh,c^2=ef,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323988);