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×#62832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
41310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 7 generators on 18 points:
(1,2);(4,7)(8,10)(11,12)(13,14)(15,16)(17,18);(5,9)(8,10)(12,16)(14,17);(3,4)(5,8)(6,7)(9,10)(11,12,15,16)(13,17,18,14);(11,13)(12,17)(14,16)(15,18);(3,5,6,9)(4,8,7,10)(11,14,15,17)(12,13,16,18);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,16)(15,18)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493377);