Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#41377 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(7,8,12,14)(9,15,16,11)(10,13,20,22)(17,21,18,19)(23,24,27,30)(25,31,33,28)(26,29,34,36)(32,37,38,35);(9,16)(11,15)(17,18)(19,21)(23,25,27,33)(24,28,30,31)(26,32,34,38)(29,35,36,37);(3,4)(5,6)(7,9)(8,11)(10,18)(12,16)(13,21)(14,15)(17,20)(19,22)(23,25)(24,28)(26,32)(27,33)(29,35)(30,31)(34,38)(36,37);(7,10)(8,13)(9,17)(11,19)(12,20)(14,22)(15,21)(16,18)(23,24)(25,28)(26,29)(27,30)(31,33)(32,35)(34,36)(37,38);(7,11)(8,9)(10,19)(12,15)(13,17)(14,16)(18,22)(20,21)(23,26,27,34)(24,29,30,36)(25,32,33,38)(28,35,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547168);