Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
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 | |
527 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,8)(7,10,11,12)(13,14)(15,16)(17,18)(19,20)(22,23)(24,28);(6,8)(10,12)(14,18)(15,20)(21,22,25,23)(24,26,28,27);(5,7,9,11)(6,10,8,12)(13,15)(14,19)(16,18)(17,20)(21,23,25,22)(24,26,28,27);(1,2)(3,4)(5,8,9,6)(7,10,11,12)(13,16,17,19)(14,20,18,15)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^8|a^2=ef,b^2=fg,c^2=eg,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.
| maximal subgroup of | #125768, #126063, #126130, … |
| maximal normal subgroup of | #125768, #126063, #126130, … |
| maximal quotient of | #156498, #156553, #156572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10210);