Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C22×#1789 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
19014 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)(9,10,12,15,13,11,16,14);(9,11)(10,13)(12,14)(15,16);(6,8)152 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#20606, #72239517, C22×#32888, … |
| maximal normal subgroup of | C22×#20606, #72239517, C22×#32888, … |
| maximal quotient of | C2×#186562, C2×#186568, #72239472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277921);