Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62817 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
46370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 7 generators on 18 points:
(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,15)(14,17)(16,18);(3,5)(4,8)(6,12)(7,13)(9,16)(10,11)(14,18)(15,17);(4,9)(5,12)(8,15)(10,17)(11,16)(13,18);(4,10)(8,11)(9,17)(15,16);(5,13)(8,16)(11,15)(12,18);(4,9)(8,16)(10,17)(11,15)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493360);