Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12725 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12726 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,13)(6,17)(7,20,21,11)(8,12)(9,15,23,30)(10,19)(14,27)(16,26)(18,24,33,32)(22,34,31,28)(25,29);(3,5,8,16)(4,9,12,23)(6,14,19,29)(7,15,21,30)(10,22,17,31)(11,13,20,26)(18,28,33,34)(24,25,32,27);(3,6)(4,10,12,17)(5,14)(7,18)(8,19)(9,22,23,31)(11,24,20,32)(13,25,26,27)(15,28)(16,29)(21,33)(30,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400652);