Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19697 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19710 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,12)(6,10)(7,16)(8,11)(9,14)(13,17)(15,18)(19,20)(21,28)(22,26)(23,32)(24,27)(25,31)(29,34)(30,33);(3,5,6,13,8,15,7,14)(4,9,16,18,11,17,10,12)(21,29)(25,33)(28,34)(30,31);(3,4)(5,9)(6,16)(7,10)(8,11)(12,14)(13,18)(15,17)(19,21,23,31,24,29,22,30)(20,25,26,28,27,33,32,34);(3,6,8,7)(4,10,11,16)(5,13,15,14)(9,12,17,18)(19,22,24,23)(20,26,27,32)(21,30,29,31)(25,28,33,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264489);