Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(10,13)(11,15)(17,18,22,25)(19,23,28,32)(20,21,24,31)(26,30,29,27);(2,5)(6,7)(10,13)(14,16)(17,19,22,28)(18,23,25,32)(20,26,24,29)(21,30,31,27);(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,26)(25,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,24)(18,21,25,31)(19,26,28,29)(23,30,32,27);(1,3,4,8)(2,6,5,7)(17,21)(18,24)(19,27)(20,25)(22,31)(23,26)(28,30)(29,32)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132692, #487442, #488074, … |
| maximal normal subgroup of | #132692, #487442, #488074, … |
| maximal quotient of | #585534, #586164, #587646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51378);