#409642

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

1052 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=i,d^2=efi,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, 316838);