Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#21077 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,6,5)(7,8)(9,18)(10,22)(11,16)(12,27)(13,17)(14,20)(15,25)(19,33)(21,29)(23,37)(24,31)(26,30)(28,38)(32,35)(34,36);(3,5,6,4)(7,9,12,20,13,21,11,26)(8,14,16,18,17,30,27,29)(10,19,24,34,25,35,23,38)(15,28,31,32,22,36,37,33);(7,10,12,24,13,25,11,23)(8,15,16,31,17,22,27,37)(9,19,20,34,21,35,26,38)(14,28,18,32,30,36,29,33);(7,11,13,12)(8,16,17,27)(9,20,21,26)(10,23,25,24)(14,29,30,18)(15,31,22,37)(19,34,35,38)(28,33,36,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278863);