#411468

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

866 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ehi,c^2=ei,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,f^c=fi

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, 318664);