Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#547 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
524 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,7)(8,9)(11,12)(13,14,16,18,17,20,15,19);(2,4)(5,6,10,12)(7,8,11,9)(14,18)(15,16)(19,20);(5,7)(6,8)(9,12)(10,11)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8446, #186601, #186610, … |
| maximal normal subgroup of | C2×#8446, #186601, #186610, … |
| maximal quotient of | #97488, #97497, #97501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1434);