Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C4×#1905 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1835 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);(5,6,10,14,11,16,9,15)(7,17,20,27,21,30,19,12)(8,22,24,13,25,28,23,29)(18,26,33,35,34,36,32,31);(5,7)(6,12,16,27)(8,18)(9,19)(10,20)(11,21)(13,26,29,36)(14,17,15,30)(22,31,28,35)(23,32)(24,33)(25,34);(5,8,11,25)(6,13,16,29)(7,18,21,34)(9,24,10,23)(12,26,27,36)(14,22,15,28)(17,31,30,35)(19,33,20,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261445);