Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#2070 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #396 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#396 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,11,14,10,15,9,23)(7,16,25,27,19,32,18,12)(8,13,24,28,22,29,21,30)(17,31,34,26,20,35,33,36);(5,7,9,18,10,19,11,25)(6,12,14,16,15,27,23,32)(8,20,21,34,22,17,24,33)(13,26,28,35,29,36,30,31);(5,8,10,22)(6,13,15,29)(7,17,19,20)(9,24,11,21)(12,26,27,36)(14,30,23,28)(16,31,32,35)(18,34,25,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266585);