Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31420 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,14)(6,18)(7,11)(8,13)(9,16)(10,19)(12,23)(15,30)(17,26)(20,29)(21,25)(22,28)(24,33)(27,32)(31,34);(3,5,7,21,8,16,23,17)(4,9,12,25,13,14,11,26)(6,15,19,33,20,31,22,32)(10,27,18,34,28,24,29,30);(3,6,8,20)(4,10,13,28)(5,15)(7,22,23,19)(9,24)(11,18,12,29)(14,27)(16,31)(17,33)(21,32)(25,34)(26,30);(4,11)(5,16)(7,23)(9,25)(10,18)(12,13)(14,26)(15,31)(19,22)(24,34)(27,30)(28,29)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411868);