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:
(1,2,5,7)(3,4,8,6)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,30,29)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,3,5,8)(2,6,7,4)(10,14)(12,21)(15,17)(18,24)(27,34)(31,33)(35,40)(38,39);(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28,30,37)(26,32,29,36)(27,35,34,40)(31,38,33,39);(1,4)(2,3)(5,6)(7,8)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29,30,26)(27,33,34,31)(28,36,37,32)(35,39,40,38)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=gh,d^2=h,e^2=g,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×#58838, #8105085, #8105135, … |
| maximal normal subgroup of | C2×#58838, #8105085, #8105135, … |
| maximal quotient of | #585717, #586021, #586517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51826);