#6578665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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	#525	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485861);