Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C4×#2424 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2424 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4);(5,6)(7,8)(13,14,16,15);(6,8)(9,10)(11,12)(13,15,16,14);(10,12)(14,15);(6,8)152 conjugacy classes — view the full character table.
| maximal subgroup of | #72239620, #72261333, #72261351, … |
| maximal normal subgroup of | #72239620, #72261333, #72261351, … |
| maximal quotient of | #72239687, #72298507, C4×#14722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 422057);