#6497964

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^d=ei

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, 6405160);