Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11065 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2710 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)(7,8,12,16,13,18,11,17)(9,14,21,29,22,30,20,34)(10,23,25,15,26,32,27,33)(19,31,35,28,36,38,24,37);(3,4,5,6)(7,9)(8,14)(10,24)(11,20)(12,21)(13,22)(15,31)(16,29)(17,34)(18,30)(19,25)(23,37)(26,35)(27,36)(28,32)(33,38);(7,10)(8,15)(9,19)(11,27)(12,25)(13,26)(14,28)(16,32)(17,23)(18,33)(20,24)(21,35)(22,36)(29,38)(30,37)(31,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264918);