#6685528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
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
prime radical	#60773	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1761 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,26)(13,22)(14,20)(15,21)(16,19)(17,25)(23,32)(24,34)(27,36)(28,38)(29,40)(30,37)(31,35)(33,39);(2,5)(3,7)(9,11,15,25)(10,17,21,18)(12,16,23,13)(14,29,30,27)(19,26,22,32)(20,36,37,40)(24,31,39,28)(33,35,34,38);(9,12,16,11,15,23,13,25)(10,18,22,26,21,17,19,32)(14,27,31,24,30,29,28,39)(20,34,38,36,37,33,35,40);(2,5)(6,8)(9,13,15,16)(10,19,21,22)(11,23,25,12)(14,28,30,31)(17,32,18,26)(20,35,37,38)(24,29,39,27)(33,40,34,36);(1,3,4,7)(2,6,5,8)(9,14,15,30)(10,20,21,37)(11,24,25,39)(12,27,23,29)(13,28,16,31)(17,33,18,34)(19,35,22,38)(26,36,32,40)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6592724);