Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#548 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
585 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 585 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,13,17,14,18,12,16)(11,15,19,22,20,23,21,24);(2,5)(3,7);(9,11,14,20)(10,15,18,23)(12,21,13,19)(16,24,17,22);(9,12,14,13)(10,16,18,17)(11,19,20,21)(15,22,23,24)Polycyclic presentation:
2^8|a^2=f,c^2=gh,d^2=gh,f^2=h,b^a=be,e^a=eg,f^b=fg,d^c=dgh55 conjugacy classes — view the full character table.
| Fitting subgroup of | #17425⋊C3 |
| maximal subgroup of | #138793, #138795, #138809, … |
| maximal normal subgroup of | #138793, #138795, #138809, … |
| maximal quotient of | #164339, #164341, #164365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10413);