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:
(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,26,30,33)(27,31,28,35)(29,32,37,40)(34,38,36,39);(1,2)(3,7)(4,5)(6,8)(10,13)(12,21)(15,17)(18,22)(25,27,30,28)(26,31,33,35)(29,36,37,34)(32,39,40,38);(2,5)(6,7)(9,11)(10,15)(12,22)(13,17)(14,19)(16,24)(18,21)(20,23)(25,28,30,27)(26,31,33,35)(29,36,37,34)(32,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,24)(19,22)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,24,22,23)(25,29,30,37)(26,32,33,40)(27,34,28,36)(31,38,35,39)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fg,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×#53863, #8103464, #8103476, … |
| maximal normal subgroup of | C2×#53863, #8103464, #8103476, … |
| maximal quotient of | #586418, #587066, #587074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46851);