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 | #2703 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,26,30,33)(27,31,28,35)(29,32,37,40)(34,38,36,39);(1,2)(3,4)(5,6)(7,8)(10,14)(15,21)(16,19)(22,23)(25,27,30,28)(26,31,33,35)(29,36,37,34)(32,39,40,38);(2,6)(3,7)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28,30,27)(26,31,33,35)(29,36,37,34)(32,38,40,39);(1,3,5,7)(2,4,6,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,12,13,20)(10,16,14,19)(11,17,18,24)(15,22,21,23)(25,29,30,37)(26,32,33,40)(27,34,28,36)(31,38,35,39)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,d^2=h,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×#58161, #8104369, #8104428, … |
| maximal normal subgroup of | C2×#58161, #8104369, #8104428, … |
| maximal quotient of | #586580, #586747, #587297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51149);