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×#3541 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3541 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4);(6,9)(7,11)(13,14)(15,21)(16,18)(17,24)(19,22)(20,25)(23,26)(27,28);(6,9)(10,12)(14,16)(15,22)(20,24)(23,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,16,18,14)(15,21,22,19)(17,24,25,20)(23,27,28,26);(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533363);