#6576731

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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

2598 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)(11,15)(12,17)(13,16)(14,19)(18,29)(20,33)(21,31)(22,32)(23,30)(24,35)(25,27)(26,34)(28,36)(37,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,24)(13,20)(14,23)(16,33)(17,35)(18,32)(19,30)(21,27)(22,29)(25,31)(26,37)(28,38)(34,39)(36,40);(10,16)(11,20)(13,29)(15,30)(17,34)(18,19)(21,24)(22,23)(25,36)(26,27)(31,39)(37,38);(1,4,5,3)(2,6,7,8)(9,10)(11,15)(12,25)(13,18)(14,19)(16,29)(17,27)(20,32)(21,35)(22,33)(23,30)(24,31)(26,36)(28,34)(37,40)(38,39);(9,12,14,28)(10,17,19,36)(11,21,23,37)(13,26,29,27)(15,31,30,39)(16,34,18,25)(20,24,22,38)(32,40,33,35)

Polycyclic presentation:

2^9|b^2=f,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483927);