Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#9929 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,12,15)(9,16,18,21)(10,14,11,19)(13,20,22,17)(23,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,37)(36,38);(7,9,12,18)(8,13,15,22)(10,17,11,20)(14,21,19,16)(23,25,27,32)(24,28,30,35)(26,31,33,38)(29,36,37,34);(4,6)(7,10,12,11)(8,14,15,19)(9,17,18,20)(13,21,22,16)(23,26)(24,29)(25,31)(27,33)(28,34)(30,37)(32,38)(35,36)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141663);