Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×#1521 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1220 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)(7,13,15,10)(8,11)(9,12)(14,19,16,18)(17,20)(21,22)(23,29)(24,32)(25,27)(26,31)(28,34)(30,36)(33,35);(5,7)(6,10)(8,16)(9,15)(11,19)(12,13)(14,17)(18,20)(21,23)(22,26)(24,33)(25,30)(27,35)(28,36)(29,32)(31,34);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,27)(23,30)(28,32)(31,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89837);