Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1445 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16,19,27)(8,20)(9,14)(10,23)(11,15)(12,18,28,24)(13,21)(17,29,35,26)(22,30)(25,31)(32,33,36,34);(1,2)(3,4)(5,7)(6,12,15,28)(8,17)(9,18)(10,24)(11,19)(13,29,31,26)(14,16,23,27)(20,32,30,36)(21,33)(22,35)(25,34);(2,4)(5,8,9,21,11,22,10,25)(6,13,14,30,15,31,23,20)(7,17,18,33,19,35,24,34)(12,26,16,32,28,29,27,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30868);