Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#400 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C23 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1775 conjugacy classes of subgroups; only the 823 normal subgroups are shown.
This group has 1775 conjugacy classes of subgroups — too many to draw. Only its 823 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,4);(1,2)(3,4);(11,15)(13,16)(17,18,20,22)(19,23,24,21);(6,8)(9,10,12,14)(11,16,15,13)(17,18)(19,23)(20,22)(21,24);(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,20,24)(18,21,22,23)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#400 |
| maximal subgroup of | #97086, #97088, #97092, … |
| maximal normal subgroup of | #97086, #97088, #97092, … |
| maximal quotient of | C22×#1441, C22×#1442, C22×#1443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27529);