#470911

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

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

828 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=ef,d^2=e,e^2=i,f^2=i,g^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,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi

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, 378107);