Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
367 conjugacy classes of subgroups; only the 265 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 265 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,5,4,6,7,8)(10,14)(12,15)(16,20)(17,21)(19,22)(23,24);(9,10,15,21,13,17,12,14)(11,16,22,24,18,23,19,20);(9,11)(10,16)(12,19)(13,18)(14,20)(15,22)(17,23)(21,24);(11,18)(16,23)(19,22)(20,24)Polycyclic presentation:
2^8|a^2=f,b^2=eg,e^2=g,f^2=h,b^a=be,e^a=eg,d^c=dg88 conjugacy classes — view the full character table.
| Fitting subgroup of | #17789⋊C3 |
| maximal subgroup of | #135820, #135821, #136607, … |
| maximal normal subgroup of | #135820, #135821, #136607, … |
| maximal quotient of | #180912, #183366, #183576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10777);