Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#48829 |
| 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 | |
6766 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)(7,8,12,15)(9,10,13,17)(11,14,19,21)(16,18,20,22)(23,24,28,29)(25,30,33,37)(26,31,32,27)(34,35,36,38);(9,13)(10,17)(11,19)(14,21)(24,29)(25,33)(27,31)(34,36);(3,4,5,6)(7,9)(8,10)(11,16)(12,13)(14,18)(15,17)(19,20)(21,22)(23,25)(24,30)(26,36)(27,35)(28,33)(29,37)(31,38)(32,34);(7,10)(8,13)(9,15)(11,18)(12,17)(14,20)(16,21)(19,22)(23,26,28,32)(24,31,29,27)(25,34,33,36)(30,35,37,38);(7,11)(8,14)(9,16)(10,18)(12,19)(13,20)(15,21)(17,22)(23,27,28,31)(24,32,29,26)(25,35,33,38)(30,34,37,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611943);