Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2538 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);(7,8,12,18,13,16,11,17)(9,19,22,29,23,14,21,30)(10,24,25,32,26,33,27,15)(20,34,35,28,36,38,37,31);(3,4)(5,6)(7,9,13,23)(8,14,16,19)(10,20,26,36)(11,22,12,21)(15,31,32,28)(17,30,18,29)(24,38,33,34)(25,37,27,35);(4,6)(7,10)(8,15,16,32)(9,20)(11,25)(12,27)(13,26)(14,28,19,31)(17,24,18,33)(21,35)(22,37)(23,36)(29,38,30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265074);