Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
165 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 53 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 — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,16,14,17,22,13)(11,15,19,23,21,24,20,18);(2,5)(6,8)(9,11)(10,15)(12,19)(13,18)(14,21)(16,23)(17,24)(20,22);(1,3,4,7)(2,6,5,8)(10,13)(11,18)(12,22)(15,20)(16,17)(19,24)(21,23)Polycyclic presentation:
2^8|a^2=e,c^2=d,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,c^b=cf,f^b=fg,g^b=gh,e^c=eh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #126279, #126280, #126281, … |
| maximal normal subgroup of | #126279, #126280, #126281, … |
| maximal quotient of | #146004, #146005, #146017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6472);