Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#60300 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C2×#3475 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20,24,28,23,25,22,27)(21,29,30,34,31,26,32,33);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(20,25)(26,29)(27,28)(33,34);(5,12)(8,11)(13,18)(15,17)(20,25)(26,29)(27,28)(33,34);(4,9)(8,11)(10,16)(15,17)(19,21)(20,26)(22,32)(23,31)(24,30)(25,29)(27,34)(28,33);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)(19,22,23,24)(20,27,25,28)(21,30,31,32)(26,33,29,34)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481603);