#6467865

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 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)(11,20)(12,17)(13,28)(14,18)(15,19)(16,22)(21,36)(23,31)(24,30)(25,34)(26,39)(27,35)(29,32)(33,37)(38,40);(9,11,13,22,15,23,14,29)(10,16,18,20,19,32,28,31)(12,24,26,37,25,38,27,21)(17,33,35,30,34,36,39,40);(2,5)(6,8)(9,12,14,27,15,25,13,26)(10,17,18,35,19,34,28,39)(11,21,22,24,23,37,29,38)(16,30,31,33,32,40,20,36);(9,12,13,26,15,25,14,27)(10,17,28,39,19,34,18,35)(11,21,29,38,23,37,22,24)(16,30,20,36,32,40,31,33);(1,3,4,7)(2,6,5,8)(12,25)(17,34)(21,37)(24,38)(26,27)(30,40)(33,36)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375061);