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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26,30,29)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,2,5,3)(4,7,8,6)(10,12)(15,17)(16,20)(22,23)(27,34)(31,33)(35,40)(38,39);(2,3)(6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,34)(26,31,29,33)(28,35,37,40)(32,38,36,39);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,37)(26,32,29,36)(27,35,34,40)(31,38,33,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,26)(27,33,34,31)(28,36,37,32)(35,39,40,38)Polycyclic presentation:
2^8|a^2=fg,b^2=gh,c^2=fg,d^2=fh,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×#58367, #8104111, #8104120, … |
| maximal normal subgroup of | C2×#58367, #8104111, #8104120, … |
| maximal quotient of | #586692, #587058, #587292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51355);