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)(9,10,16,14)(11,17,13,12)(15,18,23,22)(19,24,21,20);(1,2)(3,6)(4,7)(5,8)(9,11,16,13)(10,12,14,17)(15,19,23,21)(18,20,22,24);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,14)(15,20)(16,17)(18,21)(19,22)(23,24);(9,13,16,11)(10,17,14,12)(15,21,23,19)(18,24,22,20);(1,3)(2,6)(4,8)(5,7)(9,14,16,10)(11,17,13,12)(15,18,23,22)(19,20,21,24);(1,2)(3,5)(4,7)(6,8)(9,15)(10,18)(11,19)(12,20)(13,21)(14,22)(16,23)(17,24)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh76 conjugacy classes — view the full character table.
| Fitting subgroup of | #63020⋊C3 |
| maximal subgroup of | #493644, #511822, #7622884, … |
| maximal normal subgroup of | #493644, #511822, #7622884, … |
| maximal quotient of | #7627220, #7627228, #7627240, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56008);