Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#2823 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1251 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1251 conjugacy classes of subgroups — too many to draw. Only its 487 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(7,8,10,12)(9,11,13,14)(15,16,19,20)(17,18,21,22);(4,6)(9,13)(11,14)(16,20)(18,22);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,17)(16,18)(19,21)(20,22);(7,9,10,13)(8,11,12,14)(15,18)(16,21)(17,20)(19,22)52 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2823⋊C3 |
| maximal subgroup of | #131379, C2×#12155, C2×#12249, … |
| maximal normal subgroup of | #131379, C2×#12155, C2×#12249, … |
| maximal quotient of | C2×#14632, C2×#14633, C2×#14634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29670);