Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#1440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C26 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2311 conjugacy classes of subgroups; only the 559 normal subgroups are shown.
This group has 2311 conjugacy classes of subgroups — too many to draw. Only its 559 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) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(8,10)(11,12,13,14)(15,16,18,17);(4,6)(8,10)(16,17);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17,18,16)88 conjugacy classes — view the full character table.
| maximal subgroup of | #108223, #115349, C2×#7324, … |
| maximal normal subgroup of | #108223, #115349, C2×#7324, … |
| maximal quotient of | C2×#7555, C2×#7556, C4×#1440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6825);