#415548

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 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)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,21)(22,24)(25,26,29,34,30,33,28,32)(27,35,37,31,38,39,36,40);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,21)(22,23)(25,27,30,38)(26,31,33,40)(28,37,29,36)(32,39,34,35);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(18,21)(22,23)(26,32)(28,29)(31,39)(33,34)(35,40)(36,37);(17,19,20,24)(18,22,21,23)(26,33)(31,40)(32,34)(35,39)

Polycyclic presentation:

2^9|a^2=egh,b^2=efi,c^2=ef,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322744);