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×#41411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6954 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)(7,8,12,13)(9,15,17,11)(10,18,21,14)(16,20,19,22)(23,24,28,32)(25,33,34,29)(26,30,36,38)(27,31,37,35);(8,13)(11,15)(14,18)(20,22)(25,34)(27,37)(29,33)(31,35);(3,4,5,6)(7,9)(8,11)(10,19)(12,17)(13,15)(14,22)(16,21)(18,20)(23,25,28,34)(24,29,32,33)(26,35,36,31)(27,30,37,38);(7,10,12,21)(8,14,13,18)(9,16,17,19)(11,20,15,22)(23,26)(24,30)(25,31)(27,29)(28,36)(32,38)(33,37)(34,35);(7,11)(8,9)(10,20)(12,15)(13,17)(14,16)(18,19)(21,22)(23,27)(24,31)(25,30)(26,29)(28,37)(32,35)(33,36)(34,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546981);