Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C4×#3553 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3553 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 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)(6,14)(7,15)(8,16)(10,18)(11,17)(12,19)(13,20)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533393);