#6584325

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574063
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1983 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)(3,8)(4,6)(5,7)(9,10,16,22)(11,17,12,19)(13,20,30,40)(14,31,15,21)(18,24,34,26)(23,33,25,36)(27,35,28,37)(29,39,32,38);(1,3,5,4)(2,6,7,8)(9,11,16,12)(10,17,22,19)(13,27,30,28)(14,24,15,26)(18,31,34,21)(20,35,40,37)(23,32,25,29)(33,38,36,39);(9,12,15,24,16,11,14,26)(10,18,31,17,22,34,21,19)(13,25,32,28,30,23,29,27)(20,35,39,33,40,37,38,36);(1,4,5,3)(2,6,7,8)(9,12)(10,19)(11,16)(13,28)(14,26)(15,24)(17,22)(18,31)(20,37)(21,34)(23,32)(25,29)(27,30)(33,38)(35,40)(36,39);(9,13)(10,20)(11,23)(12,25)(14,29)(15,32)(16,30)(17,33)(18,35)(19,36)(21,38)(22,40)(24,28)(26,27)(31,39)(34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491521);