Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×SD16×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C23×D4 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(3,4,5,6,7,8,9,10);(4,6)(5,9)(8,10);(11,12)(13,16)(14,15)(17,18);(12,15)(16,18);(11,13,14,17)(12,16,15,18)140 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10576982, C2×#10577064, C2×#10577127, … |
| maximal normal subgroup of | C2×#10576982, C2×#10577064, C2×#10577127, … |
| maximal quotient of | C2×#6596240, SD16×#525, #72305462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483963);