Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,8)(10,15)(11,19)(12,22)(18,23);(3,7)(6,8)(9,10,14,15)(11,20,19,16)(12,17,22,24)(13,23,21,18);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,16,15,20)(12,21,22,13)(17,23,24,18);(9,12)(10,17)(11,21)(13,19)(14,22)(15,24)(16,23)(18,20);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,18,15,23)(11,12,19,22)(16,17,20,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,21)(13,19)(14,22)(15,24)(16,23)(18,20)Polycyclic presentation:
2^8|b^2=g,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #186328, #186330, #186331, … |
| maximal normal subgroup of | #186328, #186330, #186331, … |
| maximal quotient of | #7626231, #7626242, #7626265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55821);