Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1574 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);(3,4,9,7)(5,12,14,18)(6,11,8,15)(10,17,16,13)(19,20)(21,28)(22,26)(23,31)(24,27)(25,30)(29,34)(32,33);(3,5,9,14)(4,10,7,16)(6,13,8,17)(11,18,15,12)(21,29)(25,33)(28,34)(30,32);(3,6,9,8)(4,11,7,15)(5,13,14,17)(10,18,16,12)(19,21)(20,25)(22,30)(23,32)(24,29)(26,34)(27,33)(28,31);(3,7,9,4)(5,12,14,18)(6,15,8,11)(10,17,16,13)(19,22,24,23)(20,26,27,31)(21,30,29,32)(25,34,33,28)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266657);