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 | |
691 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)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(4,8)(6,7)(10,13)(12,21)(15,17)(18,22)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(2,3)(6,7)(9,11)(10,15)(12,22)(13,17)(14,19)(16,24)(18,21)(20,23)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,24)(19,22)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,24,22,23)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^2=fg,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×#57618, #8101833, #8101947, … |
| maximal normal subgroup of | C2×#57618, #8101833, #8101947, … |
| maximal quotient of | #586008, #586325, #586861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50606);