Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#2315 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C26 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2591 conjugacy classes of subgroups; only the 1039 normal subgroups are shown.
This group has 2591 conjugacy classes of subgroups — too many to draw. Only its 1039 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,6)(7,8,9,10)(11,12)(13,15)(14,17)(16,18);(12,13)(16,17);(3,4,5,6)(11,13,15,12)(14,17,18,16);(11,14)(12,16)(13,17)(15,18)88 conjugacy classes — view the full character table.
| maximal subgroup of | #105811, #106812, #106834, … |
| maximal normal subgroup of | #105811, #106812, #106834, … |
| maximal quotient of | C2×#13778, C2×#13782, C4×#2315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27037);