Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30871 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#30873 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,9,15)(5,16,21,30)(6,7,12,27)(8,14,28,13)(10,20,31,19)(11,23,26,24)(17,25,33,29)(18,22,34,32);(3,5)(4,10)(6,22)(7,25)(8,20)(9,21)(11,18)(12,32)(13,30)(14,16)(15,31)(17,24)(19,28)(23,33)(26,34)(27,29);(3,6)(4,11)(5,17)(7,14)(8,23)(9,12)(10,29)(13,27)(15,26)(16,18)(19,32)(20,22)(21,33)(24,28)(25,31)(30,34);(3,7,8,11,9,27,28,26)(4,12,13,24,15,6,14,23)(5,18,19,25,21,34,20,29)(10,17,16,32,31,33,30,22)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409358);