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) = C26×C4 |
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 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(1,3,4,8)(2,5,7,6)(10,14)(11,18)(16,20)(19,24)(26,29)(28,37)(31,33)(34,38);(3,8)(5,6)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27)(26,31)(28,38)(29,33)(30,35)(32,40)(34,37)(36,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,39)(33,40)(35,38);(1,2)(3,6)(4,7)(5,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,40,38,39)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=fgh,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×#56959, #8099664, #8100102, … |
| maximal normal subgroup of | C2×#56959, #8099664, #8100102, … |
| maximal quotient of | #586261, #586351, #587285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49947);