Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#25074 |
| 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 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1778 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)(13,15)(14,16)(17,18)(19,20,22,24)(21,27,29,25)(23,26,30,32)(28,33,31,34);(3,5,7,6)(4,8,10,9)(11,13,14,17)(12,15,16,18)(20,24)(21,28)(23,30)(25,33)(27,34)(29,31);(12,16)(15,18)(19,21,22,29)(20,25,24,27)(23,31,30,28)(26,33,32,34);(3,6,7,5)(4,9,10,8)(20,24)(21,29)(26,32)(28,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353208);