Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13023 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8,12,18,27,36,11,15,14,20,10,17,13,19,28,32)(9,21,24,16,30,31,29,38,26,37,22,35,25,33,23,34);(3,4,5,6)(8,15)(10,11)(12,28)(13,27)(16,31)(17,19)(18,36)(20,32)(21,38)(22,29)(23,24)(25,30)(33,35)(34,37);(7,9)(8,16)(10,23)(11,24)(12,29)(13,30)(14,26)(15,31)(17,21)(18,37)(19,38)(20,35)(22,28)(25,27)(32,33)(34,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401721);