#6687002

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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

1848 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)(9,10,14,13)(11,20,23,16)(12,26,28,36)(15,19,32,18)(17,31,29,30)(21,35,38,40)(22,24,34,25)(27,37,39,33);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,27)(13,20)(14,23)(15,24)(17,35)(18,34)(19,22)(21,31)(25,32)(26,33)(28,39)(29,40)(30,38)(36,37);(9,12,15,30,14,28,32,31)(10,17,18,26,13,29,19,36)(11,21,24,27,23,38,25,39)(16,33,34,40,20,37,22,35);(1,3,5,4)(2,6,7,8)(10,13)(12,28)(16,20)(18,19)(21,38)(22,34)(27,39)(30,31);(1,4,5,3)(2,6,7,8)(9,13,14,10)(11,22,23,34)(12,29,28,17)(15,19,32,18)(16,25,20,24)(21,35,38,40)(26,30,36,31)(27,33,39,37)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=fi,e^2=f,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,e^d=ei,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, 6594198);