Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#20678 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×#121 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,12,17,13,18,11,16)(9,19,22,30,23,29,21,14)(10,24,25,15,27,31,26,32)(20,34,35,33,37,28,36,38);(7,9,11,21,13,23,12,22)(8,14,17,19,18,30,16,29)(10,20,25,35,27,37,26,36)(15,28,24,33,32,34,31,38);(4,6)(7,10)(8,15)(9,20)(11,25)(12,26)(13,27)(14,28)(16,31)(17,24)(18,32)(19,33)(21,35)(22,36)(23,37)(29,38)(30,34);(7,11,13,12)(8,16,18,17)(9,21,23,22)(10,25,27,26)(14,29,30,19)(15,31,32,24)(20,35,37,36)(28,38,34,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278027);