Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
611 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 611 conjugacy classes of subgroups — too many to draw. Only its 175 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) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,8)(13,14,17,19)(15,20,18,16);(5,6,8,7)(10,12)(15,18)(16,20);(6,7)(9,10)(11,12)(13,15,17,18)(14,16,19,20);(1,2)(3,4)(5,7,8,6)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^8|a^2=ef,b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| Fitting subgroup of | #17114⋊C3, #72267837, #72267877 |
| maximal subgroup of | #125767, #126062, #126132, … |
| maximal normal subgroup of | #125767, #126062, #126132, … |
| maximal quotient of | #156482, #156577, #156660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10102);