Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(26,27)(28,35)(29,37)(38,39);(4,8)(6,7)(11,19)(13,23)(15,24)(17,20)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39);(2,3)(6,7)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=g,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 | C2×#54133, #8103787, #8103888, … |
| maximal normal subgroup of | C2×#54133, #8103787, #8103888, … |
| maximal quotient of | #585732, #586218, #586830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47121);