#6820010

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1161 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fgh,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727206);