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 | |
3631 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)(10,15)(11,17)(18,23)(19,24);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24);(1,2)(3,5)(4,7)(6,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)Polycyclic presentation:
2^8|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 | #63019⋊C3 |
| maximal subgroup of | #6350243, #6350244, #6350257, … |
| maximal normal subgroup of | #6350243, #6350244, #6350257, … |
| maximal quotient of | #7627219, #7627229, #7627241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56007);