Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#7779 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(7,10)(12,15)(13,17)(16,18)(19,20,24,27)(21,28,30,33)(22,26,23,31)(25,32,34,29);(3,4,6,8)(5,7,9,10)(13,17)(15,18)(19,21)(20,25)(22,29)(23,32)(24,30)(26,33)(27,34)(28,31);(3,5)(4,7)(6,9)(8,10)(11,12)(13,18)(14,16)(15,17)(19,22,24,23)(20,26,27,31)(21,29,30,32)(25,33,34,28)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 72972);