Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7813 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(4,8)(5,6)(9,10)(12,15)(13,17)(16,18)(19,20,23,26)(21,24,28,31)(22,25,29,32)(27,33,34,30);(3,4,7,9)(5,8,6,10)(19,21)(20,24)(22,27)(23,28)(25,30)(26,31)(29,34)(32,33);(3,5,7,6)(4,8,9,10)(11,12,14,16)(13,18,17,15)(19,22)(20,25)(21,27)(23,29)(24,30)(26,32)(28,34)(31,33)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 72966);