Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C22×#479 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1071 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1071 conjugacy classes of subgroups — too many to draw. Only its 567 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2);(3,4);(5,6,8,10)(7,11,12,9)(13,14,16,18)(15,21,22,28)(17,25,24,20)(19,26,27,23);(5,7,8,12)(6,9,10,11)(14,18)(19,27)(20,25)(21,28);(13,15,16,22)(14,19,18,27)(17,26,24,23)(20,28,25,21)76 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#479 |
| maximal subgroup of | #114516, #114729, C2×#9246, … |
| maximal normal subgroup of | #114516, #114729, C2×#9246, … |
| maximal quotient of | C22×#1805, C22×#1863, C22×#1865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53466);