Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#395 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
971 conjugacy classes of subgroups; only the 319 normal subgroups are shown.
This group has 971 conjugacy classes of subgroups — too many to draw. Only its 319 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(5,6)(7,8)(13,14,17,19)(15,20,16,18);(9,10)(11,12)(13,15,17,16)(14,18,19,20);(6,8)(10,12)(13,16,17,15)(14,18,19,20)88 conjugacy classes — view the full character table.
| Fitting subgroup of | C4×#395⋊C3 |
| maximal subgroup of | #99548, #99549, #99564, … |
| maximal normal subgroup of | #99548, #99549, #99564, … |
| maximal quotient of | C4×#1441, C4×#1446, C4×#1444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6848);