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 | |
687 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26,30,28)(27,31,33,35)(29,36,37,32)(34,40,39,38);(1,2,4,6)(3,5,8,7)(10,11)(12,19)(16,18)(20,24)(26,28)(27,33)(29,37)(38,40);(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,26)(27,31,33,35)(29,36,37,32)(34,38,39,40);(1,3,4,8)(2,5,6,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,37)(26,32,28,36)(27,34,33,39)(31,38,35,40)Polycyclic presentation:
2^8|a^2=gh,b^2=fh,c^2=g,d^2=gh,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×#58569, #8104193, #8104427, … |
| maximal normal subgroup of | C2×#58569, #8104193, #8104427, … |
| maximal quotient of | #585988, #586494, #586876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51557);