Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32657 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,9)(5,6,10,8)(11,12)(13,17)(14,16)(15,18)(19,20)(21,26)(22,30)(23,25)(24,27)(28,34)(29,32)(31,33);(3,5,7,10)(4,8,9,6)(11,13)(12,15)(14,18)(16,17)(20,25)(21,28)(22,24)(26,33)(29,31)(32,34);(12,16)(15,17)(19,21,23,29)(20,26,25,32)(22,28,24,31)(27,33,30,34);(3,6,7,8)(4,5,9,10)(13,18)(15,17)(19,22,23,24)(20,27,25,30)(21,28,29,31)(26,33,32,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7418316);