Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19162 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19164 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,7)(5,9)(8,10)(11,12,14,17)(13,15,18,16);(3,4,5,7,6,8,9,10);(12,15)(13,18)(16,17);(4,8)(7,10)86 conjugacy classes — view the full character table.
| maximal subgroup of | C2×SD16×AGL(1,ℤ/8), C2×#6361602, C2×#6354077 |
| maximal normal subgroup of | C2×SD16×AGL(1,ℤ/8), C2×#6361602, C2×#6354077 |
| maximal quotient of | C2×#197726 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260830);