Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18991 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×#120 |
| prime radical | C2×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2712 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)(5,13)(6,15)(7,10)(8,11)(9,12)(14,17)(16,18)(19,20,24,27)(21,28,30,33)(22,26,23,31)(25,32,34,29);(3,5)(4,6)(7,14)(8,15)(9,16)(10,13)(11,18)(12,17)(19,21,24,30)(20,25,27,34)(22,29,23,32)(26,33,31,28);(3,6)(4,5)(7,13)(8,18)(9,17)(10,14)(11,15)(12,16)(19,20)(21,25)(22,26)(23,31)(24,27)(28,32)(29,33)(30,34);(3,7,9,8)(4,10,12,11)(5,14,16,15)(6,13,17,18)(19,22,24,23)(20,26,27,31)(21,29,30,32)(25,33,34,28)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258566);