Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C22×#2002 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
10558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(5,6)(7,8)(9,10)(11,13)(12,15)(14,16);(10,13)(11,16)(14,15);(6,8)(11,16)(13,14)104 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#23906, C2×#475764, #72213751, … |
| maximal normal subgroup of | C22×#23906, C2×#475764, #72213751, … |
| maximal quotient of | #72213353, C2×#234400, C2×#393958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338909);