Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #481 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
177 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,11,19,17,20,7,13,10,14,16,12,9,15,8,18);(1,2)(3,4)(6,12,14,19)(7,11)(8,16)(9,17)(13,15,18,20);(2,4)(6,12)(7,16)(8,11)(9,17)(13,20)(14,19)(15,18)Polycyclic presentation:
2^8|a^2=dg,b^2=deg,d^2=g,e^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dgh,e^c=egh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #126470, #126471, #126474, … |
| maximal normal subgroup of | #126470, #126471, #126474, … |
| maximal quotient of | #146196, #146197, #146204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6523);