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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20)(26,27)(28,35)(29,37)(38,39);(2,5)(6,8)(10,15)(12,19)(17,21)(20,24)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,20,19,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)Polycyclic presentation:
2^8|a^2=fgh,b^2=gh,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×#55821, #8103020, #8103185, … |
| maximal normal subgroup of | C2×#55821, #8103020, #8103185, … |
| maximal quotient of | #586233, #586772, #586928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48809);