Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19040 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2808 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);(4,9)(5,12)(6,7)(11,14)(13,15)(16,18)(19,20)(21,28)(22,26)(23,32)(24,27)(25,30)(29,33)(31,34);(3,4,8,11)(5,10,13,17)(6,14,7,9)(12,18,15,16)(19,21,24,31)(20,25,27,33)(22,30,23,29)(26,34,32,28);(3,5)(4,10)(6,12)(7,15)(8,13)(9,16)(11,17)(14,18)(19,20)(21,25)(22,26)(23,32)(24,27)(28,29)(30,34)(31,33);(4,11)(9,14)(10,17)(16,18)(19,22,24,23)(20,26,27,32)(21,29,31,30)(25,28,33,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258629);