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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(10,14)(11,17)(16,21)(19,23);(2,3)(6,7)(12,20)(16,21)(19,23)(22,24)(25,26,30,28)(27,33,35,31)(29,36,37,32)(34,38,40,39);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(1,4,5,8)(2,6,3,7)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=fh,e^2=fg,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×#58762, #8104959, #8104991, … |
| maximal normal subgroup of | C2×#58762, #8104959, #8104991, … |
| maximal quotient of | #585716, #586390, #586717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51750);