Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(11,15)(13,16)(17,18,22,26)(19,25,21,23)(20,28,30,24)(27,31,29,32);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(17,19,22,21)(18,23,26,25)(20,29,30,27)(24,32,28,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,30)(29,32);(1,2)(3,6)(4,5)(7,8)(17,20,22,30)(18,24,26,28)(19,27,21,29)(23,31,25,32);(1,3,4,7)(2,6,5,8)(17,21,22,19)(18,25,26,23)(20,29,30,27)(24,32,28,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6497046, #6497050, #6497051, … |
| maximal normal subgroup of | #6497046, #6497050, #6497051, … |
| maximal quotient of | #553736, #553737, #554028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30279);