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 | |
3583 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)(9,10,15,20)(11,18,13,16)(12,14,19,17)(21,24,23,22);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,20,18)(12,21,19,23)(14,22,17,24);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,23)(14,20)(15,19)(16,24)(18,22);(9,13,15,11)(10,18,20,16)(12,23,19,21)(14,24,17,22);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,21)(13,23)(15,19)(16,22)(17,20)(18,24);(1,2)(3,5)(4,7)(6,8)(9,14)(10,19)(11,22)(12,20)(13,24)(15,17)(16,23)(18,21)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^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 | #63015⋊C3 |
| maximal subgroup of | #6350197, #6350198, #6350221, … |
| maximal normal subgroup of | #6350197, #6350198, #6350221, … |
| maximal quotient of | #7627215, #7627224, #7627228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56003);