Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1325 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1325 conjugacy classes of subgroups — too many to draw. Only its 575 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 — 5 generators on 20 points:
(2,4)(7,11)(10,12)(13,14,16,18)(15,19,20,17);(1,2,3,4)(6,9)(10,12);(5,6)(7,10)(8,9)(11,12)(13,15,16,20)(14,17,18,19);(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,11)(9,12)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=g,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #97130, #111300, #111465, … |
| maximal normal subgroup of | #97130, #111300, #111465, … |
| maximal quotient of | #516456, #516634, #516877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27999);