Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D16 |
| commutator | C16 | C42 |
| Frattini, cosocle | C22×C16 | C22 |
| socle | C22 | #366 |
| prime radical | C23 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2)(3,4)(5,6,9,16,13,19,12,37)(7,20,22,50,26,51,25,61)(8,17,28,45,30,47,29,62)(10,15,31,43,34,44,33,46)(11,35,32,48,36,63,27,18)(14,38,39,54,42,53,64,21)(23,49,55,66,58,67,57,41)(24,59,56,65,60,68,52,40);(2,4)(5,7,10,23,11,24,28,38,9,22,31,55,32,56,30,54,13,26,34,58,36,60,29,53,12,25,33,57,27,52,8,21)(6,14,17,40,18,41,46,61,37,64,62,68,63,67,44,51,19,42,47,65,48,66,43,50,16,39,45,59,35,49,15,20)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=f,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fg,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #94692, #94693, C2×#7472, … |
| maximal normal subgroup of | #94692, #94693, C2×#7472, … |
| maximal quotient of | #94021, #94045, #94204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 460);