Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| 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 | |
3636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,19)(12,14)(13,16)(15,18)(17,23)(20,24)(21,22);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,18,19)(12,20,23,22)(14,21,17,24);(1,3)(2,5)(4,8)(6,7)(9,11,15,13)(10,16,18,19)(12,20,23,22)(14,21,17,24);(9,12)(10,17)(11,20)(13,22)(14,18)(15,23)(16,24)(19,21);(1,3)(2,6)(4,8)(5,7)(9,13,15,11)(10,16,18,19)(12,22,23,20)(14,21,17,24);(1,2)(3,5)(4,7)(6,8)(9,14,15,17)(10,12,18,23)(11,21,13,24)(16,20,19,22)Polycyclic presentation:
2^8|b^2=g,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #346936, #347193, #347212, … |
| maximal normal subgroup of | #346936, #347193, #347212, … |
| maximal quotient of | #7627221, #7627222, #7627226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56009);