Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#2641 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1663 conjugacy classes of subgroups; only the 527 normal subgroups are shown.
This group has 1663 conjugacy classes of subgroups — too many to draw. Only its 527 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);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(16,19)(20,21);(3,4,5,6)(8,11)(12,13)(15,16,18,19)(17,20,22,21);(3,4)(5,6)(7,8)(9,13)(10,11)(12,14)(15,16)(17,21)(18,19)(20,22);(7,9,10,14)(8,12,11,13)(15,17,18,22)(16,20,19,21)64 conjugacy classes — view the full character table.
| maximal subgroup of | #97031, #111922, #111926, … |
| maximal normal subgroup of | #97031, #111922, #111926, … |
| maximal quotient of | C2×#14260, C2×#14272, C2×#14283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27608);