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 | |
3304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,8,6)(3,4,9,5)(7,10,15,14)(11,12,16,13)(19,26)(23,27)(28,31)(29,32);(1,3)(2,5)(4,6)(7,11)(8,9)(10,13)(12,14)(15,16)(17,18,22,25)(19,27,26,23)(20,21,24,30)(28,32,31,29);(1,4,8,5)(2,3,6,9)(7,12,15,13)(10,11,14,16)(17,19,22,26)(18,23,25,27)(20,28,24,31)(21,29,30,32);(1,5,8,4)(2,9,6,3)(7,13,15,12)(10,16,14,11);(1,6,8,2)(3,4,9,5)(7,10,15,14)(11,13,16,12)(17,20)(18,21)(19,28)(22,24)(23,29)(25,30)(26,31)(27,32);(1,7,8,15)(2,10,6,14)(3,11,9,16)(4,12,5,13)(17,21)(18,24)(19,29)(20,25)(22,30)(23,31)(26,32)(27,28)Polycyclic presentation:
2^8|a^2=g,b^2=gh,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 | #63036⋊C3 |
| maximal subgroup of | #346949, #347224, #6350217, … |
| maximal normal subgroup of | #346949, #347224, #6350217, … |
| maximal quotient of | #7627234, #7627246, #7627306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56024);