Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
995 conjugacy classes of subgroups; only the 535 normal subgroups are shown.
This group has 995 conjugacy classes of subgroups — too many to draw. Only its 535 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(2,5)(6,8)(17,19,20,23)(18,21,22,24);(10,13)(14,16);(1,3,4,7)(2,6,5,8)(18,22)(21,24);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #122010, #122161, #134177, … |
| maximal normal subgroup of | #122010, #122161, #134177, … |
| maximal quotient of | #607574, #607599, #607605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29628);