Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| 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)(7,15)(8,12,20,27)(9,21)(10,14)(11,17)(13,22)(16,30,32,28)(18,29)(19,34,35,26)(23,31,25,36)(24,33);(1,2)(3,4)(5,7)(6,11,14,24)(8,16)(9,17)(10,18)(12,25,27,23)(13,15,21,29)(19,31)(20,32)(22,33)(26,30,34,28)(35,36);(2,4)(5,8,9,19,10,20,22,35)(6,12,13,26,14,27,21,34)(7,16,17,31,18,32,33,36)(11,23,15,28,24,25,29,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30930);