#6575792

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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

2054 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,13)(14,15)(16,33)(17,20)(18,25)(19,30)(22,36)(23,38)(24,26)(27,37)(28,31)(29,32)(34,35)(39,40);(2,5)(6,8)(9,11,15,25)(10,18,14,21)(12,27,29,38)(13,23,32,37)(16,26,17,34)(19,36,28,40)(20,24,33,35)(22,30,39,31);(9,12,15,29)(10,19,14,28)(11,22,25,39)(13,20,32,33)(16,31,17,30)(18,23,21,37)(24,36,35,40)(26,38,34,27);(1,3,4,7)(2,6,5,8)(9,13)(10,12)(11,23)(14,29)(15,32)(16,28)(17,19)(18,27)(20,30)(21,38)(22,35)(24,39)(25,37)(26,40)(31,33)(34,36);(9,14)(10,15)(11,24)(12,28)(13,31)(16,33)(17,20)(18,34)(19,29)(21,26)(22,36)(23,27)(25,35)(30,32)(37,38)(39,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,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, 6482988);