Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#29153 |
| 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 | |
1922 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);(3,4,7,10)(5,9,6,8)(11,12,14,16)(13,15,17,18)(19,20,24,31,23,27,22,26)(21,25,32,34,29,28,30,33);(3,5,7,6)(4,8,10,9)(11,13,14,17)(12,15,16,18)(19,21)(20,25)(22,30)(23,29)(24,32)(26,33)(27,28)(31,34);(13,17)(15,18)(19,20,23,27)(21,28,29,25)(22,31,24,26)(30,33,32,34);(3,6,7,5)(4,9,10,8)(21,29)(25,28)(30,32)(33,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375235);