Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10,13,20,15,19,14,17)(11,16,23,31,24,29,21,32)(12,25,26,33,27,34,28,18)(22,36,35,39,38,40,37,30);(1,3)(2,6)(4,7)(5,8)(9,11,15,24)(10,16,19,29)(12,22,27,38)(13,23,14,21)(17,32,20,31)(18,30,33,39)(25,36,34,40)(26,35,28,37);(1,4,5,6)(2,3,7,8)(10,17)(11,21)(13,14)(16,29)(18,25)(19,20)(22,35)(23,24)(26,28)(30,39)(33,34)(37,38);(2,7)(3,8)(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,30)(17,25)(19,33)(20,34)(21,35)(23,37)(24,38)(29,39)(31,40)(32,36);(10,19)(16,29)(17,20)(18,33)(25,34)(30,39)(31,32)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410660);