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 | |
3669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,8)(9,10)(11,20)(12,23)(13,14)(15,16)(17,21)(18,22)(19,24);(3,7)(6,8)(10,16)(11,21)(13,19)(18,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,19)(13,22)(14,23)(15,21)(16,20)(18,24);(9,12,15,22)(10,18,16,23)(11,19,21,13)(14,17,24,20);(1,3)(2,6)(4,7)(5,8)(9,13,15,19)(10,14,16,24)(11,22,21,12)(17,23,20,18);(1,2)(3,6)(4,5)(7,8)(9,14)(10,19)(11,23)(12,17)(13,16)(15,24)(18,21)(20,22)Polycyclic presentation:
2^8|d^2=g,e^2=g,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 | #186942, #186945, #186955, … |
| maximal normal subgroup of | #186942, #186945, #186955, … |
| maximal quotient of | #7626207, #7626208, #7626250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55809);