#6467861

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	#60702	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1767 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)(12,18)(13,19)(14,31)(15,20)(16,21)(17,25)(23,39)(24,37)(26,35)(27,33)(28,40)(29,38)(30,34)(32,36);(9,11,14,25,16,26,15,32)(10,17,20,22,21,36,31,35)(12,27,28,24,13,30,29,23)(18,37,38,33,19,39,40,34);(2,5)(6,8)(9,12,15,29,16,13,14,28)(10,18,20,38,21,19,31,40)(11,23,25,27,26,24,32,30)(17,33,35,37,36,34,22,39);(9,13,15,28,16,12,14,29)(10,19,20,40,21,18,31,38)(11,24,25,30,26,23,32,27)(17,34,35,39,36,33,22,37);(1,3,4,7)(2,6,5,8)(12,13)(18,19)(23,24)(27,30)(28,29)(33,34)(37,39)(38,40)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=h,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,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, 6375057);