Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#1595 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
1020 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(15,18);(4,7)(5,10)(8,9)(11,12)(13,15)(14,16)(17,18);(11,13)(12,15)(14,17)(16,18)46 conjugacy classes — view the full character table.
| maximal subgroup of | #137355, #137357, #137363, … |
| maximal normal subgroup of | #137355, #137357, #137363, … |
| maximal quotient of | C2×#7734, C2×#7750, C2×#7735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11082);