Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C22×#2179 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C4×D8 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2);(3,4);(5,6,10,15,11,17,27,16)(7,18,20,33,22,35,21,29)(8,14,23,32,24,13,9,25)(12,28,30,36,31,26,34,19);(5,7)(6,12)(8,19)(9,26)(10,20)(11,22)(13,29)(14,33)(15,30)(16,34)(17,31)(18,25)(21,27)(23,28)(24,36)(32,35);(5,8,10,23,11,24,27,9)(6,13,15,25,17,14,16,32)(7,19,20,28,22,36,21,26)(12,29,30,18,31,33,34,35)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529818);