Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| 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 | |
3562 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)(3,4)(5,6)(7,8)(9,10)(11,19)(12,14)(13,17)(15,16)(18,22)(20,24)(21,23);(2,6)(3,7)(10,16)(12,22)(17,19)(20,23);(1,3,5,7)(2,4,6,8)(9,11,15,13)(10,17,16,19)(12,20,22,23)(14,21,18,24);(1,4)(2,7)(3,6)(5,8)(9,12)(10,18)(11,20)(13,23)(14,16)(15,22)(17,24)(19,21);(9,13,15,11)(10,17,16,19)(12,23,22,20)(14,21,18,24);(9,14,15,18)(10,12,16,22)(11,21,13,24)(17,20,19,23)Polycyclic presentation:
2^8|c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #63052⋊C3 |
| maximal subgroup of | #347441, #347461, #347483, … |
| maximal normal subgroup of | #347441, #347461, #347483, … |
| maximal quotient of | #7627409, #7627438, #7627444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56040);