Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8,11,15,12,16,10,14)(9,17,18,13,20,21,19,22)(23,24)(25,28)(26,33,34,29)(27,30)(31,35,37,38)(32,36);(3,4,5,6)(7,9)(8,13)(10,18)(11,19)(12,20)(14,21)(15,17)(16,22)(23,25,27,32)(24,28,30,36)(26,31,34,37)(29,35,33,38);(7,10,12,11)(8,14,16,15)(9,18,20,19)(13,21,22,17)(23,26)(24,29)(25,31)(27,34)(28,35)(30,33)(32,37)(36,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101903);