Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
401 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 89 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)(5,6,9,7)(8,11,10,12)(13,14,17,20)(15,18,16,19);(2,4)(6,11)(7,12)(8,10)(14,18)(15,16)(19,20);(5,7)(6,9)(8,12)(10,11)(14,19)(15,16)(18,20)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,d^c=dg,e^c=eh,f^c=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10305, #233351, #233357, … |
| maximal normal subgroup of | C2×#10305, #233351, #233357, … |
| maximal quotient of | #102786, #102787, #102793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3293);