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×#3555 |
| 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 | |
31726 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);(5,6,10,13)(7,8,15,11)(9,17,18,12)(14,19,20,16)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(5,7,10,15)(6,11,13,8)(9,14,18,20)(12,19,17,16)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,8,10,11)(6,7,13,15)(9,16,18,19)(12,14,17,20);(5,7)(6,11)(8,13)(9,14)(10,15)(12,19)(16,17)(18,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,9,10,18)(6,12,13,17)(7,14,15,20)(8,16,11,19)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493347);