Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
789 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)(3,5)(4,7)(6,8)(10,15)(11,13)(20,28)(23,30)(26,32)(29,31);(9,10,14,15)(11,12,13,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31);(2,3)(6,7)(9,11,14,13)(10,16,15,12)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,3,5,2)(4,7,8,6)(9,12,14,16)(10,11,15,13)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,16,15,12)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)Polycyclic presentation:
2^8|b^2=h,c^2=f,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #7221609, #7221613, #7221673, … |
| maximal normal subgroup of | #7221609, #7221613, #7221673, … |
| maximal quotient of | #582293, #582494, #582534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46070);