Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#21476 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,6,5)(7,8,12,16)(9,17,20,31)(10,14,21,29)(11,23,26,15)(13,24,28,19)(18,34,35,33)(22,32,25,30)(27,37,38,36);(7,9)(8,13)(10,18)(11,24)(12,20)(14,27)(15,31)(16,28)(17,23)(19,26)(21,35)(22,37)(25,36)(29,38)(30,33)(32,34);(7,10,12,21)(8,14,16,29)(9,18,20,35)(11,25,26,22)(13,27,28,38)(15,32,23,30)(17,33,31,34)(19,37,24,36);(3,5,6,4)(10,21)(14,29)(18,35)(22,25)(27,38)(30,32)(33,34)(36,37)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279897);