Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#10918 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,6,5)(15,18);(7,8,9,10)(11,12)(13,15)(14,16)(17,18);(3,5,6,4)(11,13)(12,15)(14,17)(16,18)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#1889, C2×#368021, C2×#375865 |
| maximal normal subgroup of | C2×C4×#1889, C2×#368021, C2×#375865 |
| maximal quotient of | C2×#110029 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261174);