Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11273 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #1442 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1614 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)(5,9,14,24)(6,16)(7,20)(8,11)(10,18)(12,21)(13,27,17,32)(15,31,30,23)(19,33)(22,34,25,29)(26,28);(3,5,7,15,8,14,21,30)(4,9,12,23,11,24,20,31)(6,17,18,34,19,13,28,29)(10,25,16,32,26,22,33,27);(3,6)(4,10,11,26)(5,13,14,17)(7,18)(8,19)(9,22)(12,16,20,33)(15,29,30,34)(21,28)(23,27)(24,25)(31,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265477);