Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62228 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,7,10)(5,11,12,8)(6,14,15,9)(13,16,18,17)(20,25)(21,29)(22,24)(26,33)(30,31)(32,34);(3,5)(4,8)(6,13)(7,12)(9,16)(10,11)(14,17)(15,18)(19,20)(21,26)(22,27)(23,25)(24,28)(29,34)(30,32)(31,33);(19,21)(20,26)(22,29)(23,30)(24,31)(25,32)(27,34)(28,33);(5,12)(8,11)(13,18)(16,17)(19,22,23,24)(20,27,25,28)(21,29,30,31)(26,34,32,33);(3,6,7,15)(4,9,10,14)(5,13,12,18)(8,16,11,17)(19,22,23,24)(20,28,25,27)(21,29,30,31)(26,33,32,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486677);