Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#63028 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 34 points:
(1,2);(3,4,9,7)(5,12,10,6)(8,11,16,15)(13,18,17,14)(19,20,24,27)(21,28,29,25)(22,23,26,32)(30,34,33,31);(3,5)(4,6)(7,12)(8,13)(9,10)(11,14)(15,18)(16,17)(21,29)(25,28)(30,33)(31,34);(3,5)(4,6)(7,12)(8,13)(9,10)(11,14)(15,18)(16,17)(19,21)(20,25)(22,30)(23,31)(24,29)(26,33)(27,28)(32,34);(3,6,9,12)(4,10,7,5)(8,14,16,18)(11,17,15,13);(3,7,9,4)(5,12,10,6)(8,11,16,15)(13,14,17,18)(19,22)(20,23)(21,30)(24,26)(25,31)(27,32)(28,34)(29,33);(3,8)(4,11)(5,13)(6,14)(7,15)(9,16)(10,17)(12,18)(19,23)(20,26)(21,31)(22,27)(24,32)(25,33)(28,30)(29,34)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493544);