Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
713 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 713 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,8)(9,10,13,12)(11,14,15,16)(17,18,21,24)(19,20,23,22)(26,27)(28,32);(3,7)(6,8)(10,12)(11,15)(19,23)(20,22)(25,26,29,27)(28,30,32,31);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,23)(18,22,24,20)(25,27,29,26)(28,30,32,31);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,14,15,16)(17,20,21,22)(18,23,24,19)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=f,d^2=ef,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 | #125838, #125955, #126248, … |
| maximal normal subgroup of | #125838, #125955, #126248, … |
| maximal quotient of | #156548, #156612, #156666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10261);