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×#3557 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C22 | C27 |
| prime radical | C22×#540 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
32094 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)(7,12)(8,13)(9,17)(10,11)(14,18)(15,16)(19,20)(22,26)(23,29)(24,32)(34,36);(6,11)(8,12)(14,17)(16,19)(21,22,25,26)(23,30,29,27)(24,28,32,33)(31,36,35,34);(5,7,10,13)(6,12,11,8)(9,15,18,20)(14,19,17,16)(21,22)(23,27)(24,28)(25,26)(29,30)(31,34)(32,33)(35,36);(5,8,10,12)(6,13,11,7)(9,16,18,19)(14,20,17,15)(21,23,25,29)(22,27,26,30)(24,31,32,35)(28,34,33,36);(5,9,10,18)(6,14,11,17)(7,15,13,20)(8,16,12,19)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493334);