Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#30164 |
| 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 | |
1540 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)(5,8)(6,7)(9,10)(11,12,15,17)(13,18,14,16)(19,20)(21,27)(22,24)(23,26)(25,29)(28,33)(30,32)(31,34);(11,13,15,14)(12,16,17,18)(20,24)(21,28)(23,30)(25,33)(27,34)(29,31);(4,7)(8,10)(13,14)(16,18)(19,21,22,29)(20,25,24,27)(23,31,30,28)(26,33,32,34);(3,5,6,9)(4,8,7,10)(11,14,15,13)(12,16,17,18)(20,24)(21,29)(26,32)(28,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342033);