Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C22×#441 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1395 conjugacy classes of subgroups; only the 823 normal subgroups are shown.
This group has 1395 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:
(1,2);(3,4);(5,6,8,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(5,7,8,6)(10,15)(14,23)(16,20)(17,24)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#441 |
| maximal subgroup of | #111568, #146800, C2×#9088, … |
| maximal normal subgroup of | #111568, #146800, C2×#9088, … |
| maximal quotient of | C22×#1579, C22×#1587, C22×#1593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53141);