Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214×C4 |
| direct product | C22×#2669 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
25658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2);(3,4);(5,6,7,8)(10,12);(6,8)(14,16);(9,10)(11,12)(13,14)(15,16);(5,6)(7,8)(9,10)(11,12)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1259281, C22×#52004, C22×#58990, … |
| maximal normal subgroup of | C2×#1259281, C22×#52004, C22×#58990, … |
| maximal quotient of | #72250779, C2×#7500919, C2×D3×#2669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540629);