Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#1663 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C2×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,11,15)(7,16,18,19)(8,17,12,20)(9,14,10,13)(21,22,26,24)(23,29,32,36)(25,33,35,28)(27,34,30,31);(5,7)(6,12)(8,15)(9,20)(10,17)(11,18)(13,19)(14,16)(21,23,26,32)(22,27,24,30)(25,34,35,31)(28,36,33,29);(5,8,11,12)(6,7,15,18)(9,19,10,16)(13,17,14,20)(21,24,26,22)(23,30,32,27)(25,28,35,33)(29,34,36,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116345);