#6497245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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

1348 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)(27,34)(29,30)(32,39)(33,37)(35,40)(36,38);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,26)(27,35)(28,31)(29,37)(30,33)(32,36)(34,40)(38,39);(1,3,5,4)(2,7,6,8)(11,17)(14,21)(18,23)(22,24)(25,27)(26,32)(28,35)(29,34)(30,38)(31,36)(33,40)(37,39);(1,4,5,3)(2,8,6,7)(11,17)(14,21)(18,23)(22,24)(25,28,31,26)(27,35,36,32)(29,33,30,37)(34,40,38,39);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,31,30)(26,33,28,37)(27,34,36,38)(32,40,35,39)

Polycyclic presentation:

2^9|a^2=fhi,c^2=f,d^2=fi,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,g^c=gi,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, 6404441);