#6579427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61090	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 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)(11,23)(12,19)(13,20)(14,32)(15,21)(16,22)(17,26)(18,28)(24,35)(25,38)(27,34)(29,39)(30,37)(31,36)(33,40);(2,5)(6,8)(9,11,15,31,16,13,14,26)(10,17,32,20,22,36,21,23)(12,25,33,27,29,30,28,24)(18,37,39,34,40,38,19,35);(1,3,4,7)(2,6,5,8)(9,12,14,28,16,29,15,33)(10,18,21,19,22,40,32,39)(11,24,26,30,13,27,31,25)(17,34,23,37,36,35,20,38);(9,12)(10,19)(11,25)(13,30)(14,28)(15,33)(16,29)(17,35)(18,32)(20,37)(21,40)(22,39)(23,38)(24,26)(27,31)(34,36);(9,13)(10,20)(11,16)(12,27)(14,31)(15,26)(17,21)(18,38)(19,34)(22,23)(24,29)(25,28)(30,33)(32,36)(35,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486623);