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 | #2539 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,13,15)(11,12,16,14)(17,18)(19,21)(20,24)(22,30)(23,26)(25,31)(27,32)(28,29);(2,5)(3,7)(9,11)(10,14)(12,15)(13,16)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,32,30,29);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,11,15,16)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,32,30,29);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,21,23,24)(18,19,26,20)(22,29,31,32)(25,27,30,28);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)Polycyclic presentation:
2^8|a^2=gh,b^2=fgh,c^2=f,d^2=fgh,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 | #133006, #487070, #487198, … |
| maximal normal subgroup of | #133006, #487070, #487198, … |
| maximal quotient of | #585430, #585598, #587538, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51376);