Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#33075 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,6,7)(5,8,9,10)(11,12)(13,15)(14,16)(17,18)(19,20,24,29)(21,30,25,26)(22,32,23,27)(28,33,34,31);(4,7)(8,10)(11,13,14,17)(12,15,16,18)(19,21,24,25)(20,26,29,30)(22,33,23,31)(27,34,32,28);(3,5,6,9)(4,8,7,10)(12,16)(15,18)(19,22,24,23)(20,27,29,32)(21,31,25,33)(26,28,30,34);(19,23,24,22)(20,28,29,34)(21,31,25,33)(26,27,30,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536746);