Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | C23 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,11,16)(7,17,21,48)(8,22,27,46)(9,28,31,62)(10,32,30,15)(12,34,37,18)(13,38,41,25)(14,42,44,24)(19,50,51,67)(20,52,33,47)(23,43,57,68)(26,60,59,54)(29,39,63,66)(35,61,64,56)(36,49,45,55)(40,58,53,65);(5,7,11,21)(6,12,16,37)(8,23,27,57)(9,29,31,63)(10,33,30,20)(13,39,41,66)(14,43,44,68)(15,45,32,36)(17,46,48,22)(18,38,34,25)(19,42,51,24)(26,61,59,56)(28,35,62,64)(40,67,53,50)(47,54,52,60)(49,58,55,65);(2,4)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #151174, #151175, #151180, … |
| maximal normal subgroup of | #151174, #151175, #151180, … |
| maximal quotient of | #151174, #151286, #151484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6653);