Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8,13,18,14,19,12,28)(9,20,11,15,22,17,27,31)(10,23,25,34,26,33,29,16)(21,36,37,32,38,30,24,35);(3,4)(5,6)(7,9)(8,15)(10,21)(11,13)(12,27)(14,22)(16,30)(17,18)(19,31)(20,28)(23,35)(24,29)(25,37)(26,38)(32,33)(34,36);(4,6)(7,10,14,26)(8,16,19,34)(9,21,22,38)(11,24,27,37)(12,25,13,29)(15,30,31,36)(17,32,20,35)(18,33,28,23);(7,11)(8,17)(9,12)(10,24)(13,22)(14,27)(15,28)(16,32)(18,31)(19,20)(21,25)(23,30)(26,37)(29,38)(33,36)(34,35)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345124);