Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18494 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3068 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)(7,15)(8,11)(9,13)(14,17)(16,18)(19,20)(21,27)(22,26)(23,28)(24,29)(25,30)(31,34)(32,33);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15)(19,21,25,32)(20,26,30,34)(22,23,31,24)(27,29,33,28);(19,22)(20,27)(21,23)(24,32)(25,31)(26,29)(28,34)(30,33);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(19,23,25,24)(20,28,30,29)(21,31,32,22)(26,27,34,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255233);