Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11521 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| socle | C22 | #1464 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,5,10,6,11,13,18,7,8,12,15,14,9,16,17);(4,8)(9,11);(4,9,8,11)(5,12)(10,18,15,17)(13,16)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#133259, C2×#480440, #72250353, … |
| maximal normal subgroup of | C2×#133259, C2×#480440, #72250353, … |
| maximal quotient of | C2×#128374, C2×#124849, #72219608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387508);