Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18319 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,12)(5,14,17,27)(6,18,21,11)(8,13,22,10)(9,19,25,15)(16,29,32,30)(20,26,34,28)(23,31,24,33);(3,5,7,17)(4,9,12,25)(6,19,21,15)(8,16,22,32)(10,23,13,24)(11,27,18,14)(20,33,34,31)(26,29,28,30);(6,20)(11,28)(14,29)(15,31)(18,26)(19,33)(21,34)(27,30);(4,10)(9,23)(11,26)(12,13)(14,30)(18,28)(24,25)(27,29)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254061);