Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#33194 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2988 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);(4,6)(7,8,13,18)(9,19,20,30)(10,24,26,32)(11,17,12,16)(14,23,28,22)(15,27,31,25)(21,35,34,29)(33,37,38,36);(7,9,11,22,13,20,12,23)(8,14,16,19,18,28,17,30)(10,21,25,36,26,34,27,37)(15,29,24,33,31,35,32,38);(3,4,5,6)(9,20)(14,28)(19,30)(21,34)(22,23)(29,35)(33,38)(36,37);(7,10)(8,15)(9,21)(11,25)(12,27)(13,26)(14,29)(16,24)(17,32)(18,31)(19,33)(20,34)(22,36)(23,37)(28,35)(30,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537272);