#6686993

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594189);