Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#47209 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2699 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7042 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)(9,15)(10,18)(11,12)(13,17)(14,19)(16,20)(21,22)(23,24,28,31)(25,26,29,34)(27,30,36,38)(32,33,35,37);(8,12)(13,15)(14,18)(20,21)(23,25,28,29)(24,26,31,34)(27,35,36,32)(30,37,38,33);(3,4,5,6)(7,9)(8,13)(10,16)(11,17)(12,15)(14,20)(18,21)(19,22)(23,25)(24,26)(27,32)(28,29)(30,33)(31,34)(35,36)(37,38);(7,8)(9,13)(10,14)(11,12)(15,17)(16,20)(18,19)(21,22)(23,26)(24,29)(25,31)(27,33)(28,34)(30,35)(32,38)(36,37);(7,10,11,19)(8,14,12,18)(9,16,17,22)(13,20,15,21)(23,27)(24,30)(25,32)(26,33)(28,36)(29,35)(31,38)(34,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608163);