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 | |
3618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,5)(3,8,4,7)(11,18)(15,19)(20,23)(21,24);(1,3,6,4)(2,7,5,8)(9,10,14,17)(11,19,18,15)(12,13,16,22)(20,24,23,21);(1,3,6,4)(2,7,5,8)(9,11,14,18)(10,15,17,19)(12,20,16,23)(13,21,22,24);(1,3,6,4)(2,7,5,8);(1,4,6,3)(2,7,5,8)(9,12)(10,13)(11,20)(14,16)(15,21)(17,22)(18,23)(19,24);(1,5,6,2)(3,8,4,7)(9,13)(10,16)(11,21)(12,17)(14,22)(15,23)(18,24)(19,20)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=h,d^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.
| Fitting subgroup of | #63035⋊C3 |
| maximal subgroup of | #346947, #347208, #347216, … |
| maximal normal subgroup of | #346947, #347208, #347216, … |
| maximal quotient of | #7627233, #7627244, #7627261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56023);