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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
304 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(2,4)(5,6,10,11,9,7,8,12)(13,14,16,18,17,20,15,19);(1,2,3,4)(5,7,9,6)(8,12,10,11)(14,18)(15,16)(19,20);(5,8,9,10)(6,11,7,12)(14,18)(15,16)(19,20)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8988, #193989, #193993, … |
| maximal normal subgroup of | C2×#8988, #193989, #193993, … |
| maximal quotient of | #99518, #99531, #99654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1976);