Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#2212 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#459 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6)(7,15)(8,20)(9,24)(10,26)(11,16)(12,18)(13,17)(14,25)(19,33)(21,32)(22,36)(23,28)(27,29)(30,34)(31,35);(5,7,9,18,11,19,25,10)(6,12,14,15,16,26,24,33)(8,17,21,34,23,35,22,27)(13,28,29,32,31,20,30,36);(5,8)(6,13)(7,17)(9,21)(10,27)(11,23)(12,28)(14,29)(15,32)(16,31)(18,34)(19,35)(20,26)(22,25)(24,30)(33,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403025);