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×#3533 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3533 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7318 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)(15,18)(16,21)(17,22)(19,23)(20,25)(24,26)(27,28);(6,9)(10,12)(14,15)(16,23)(20,22)(24,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533366);