Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#2006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1127 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 1127 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(8,12)(9,10)(13,14);(3,4)(5,6)(7,8)(9,14)(10,12)(11,13);(4,6)(7,8)(9,12)(10,14)(11,13)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137135, #137137, #137205, … |
| maximal normal subgroup of | #137135, #137137, #137205, … |
| maximal quotient of | C2×#9962, C2×#9963, C2×#9966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16392);