Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C22×#3563 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C22 | C27 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 36 points:
(1,2);(3,4);(5,6)(7,14)(8,11)(9,17)(10,13)(12,18)(15,19)(16,20)(21,22,27,30)(23,25,28,24)(26,34,35,29)(31,32,36,33);(5,7,10,8)(6,11,13,14)(9,15,18,16)(12,19,17,20)(21,23,27,28)(22,24,30,25)(26,31,35,36)(29,32,34,33);(5,8,10,7)(6,11,13,14)(9,16,18,15)(12,19,17,20)(21,24,27,25)(22,28,30,23)(26,32,35,33)(29,36,34,31);(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,19)(17,18)(21,25,27,24)(22,23,30,28)(26,33,35,32)(29,31,34,36);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,17)(14,20)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,36)(30,34)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493329);