#6579422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574084
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

1896 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)(11,22)(12,29)(13,19)(14,16)(15,21)(17,26)(18,31)(20,23)(24,37)(25,39)(27,34)(28,38)(30,32)(33,36)(35,40);(2,5)(6,8)(9,11,23,27,16,28,15,26)(10,17,21,38,14,34,20,22)(12,25,32,24,13,33,31,40)(18,36,19,37,30,39,29,35);(1,3,4,7)(2,6,5,8)(9,12,15,31,16,13,23,32)(10,18,20,29,14,30,21,19)(11,24,26,25,28,40,27,33)(17,35,22,39,34,37,38,36);(9,13)(10,19)(11,25)(12,16)(14,29)(15,32)(17,35)(18,20)(21,30)(22,39)(23,31)(24,27)(26,40)(28,33)(34,37)(36,38);(9,14)(10,16)(11,22)(12,30)(13,18)(15,21)(17,27)(19,31)(20,23)(24,39)(25,37)(26,34)(28,38)(29,32)(33,35)(36,40)

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,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, 6486618);