Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | C2×#121 | C22 |
| 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) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,14)(15,17)(16,21)(22,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(10,14)(12,20)(15,17)(18,24)(26,30)(27,33)(28,37)(38,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,3,4,7)(2,5,6,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^8|a^2=fgh,c^2=f,d^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46884, #7700642, #7700704, … |
| maximal normal subgroup of | C2×#46884, #7700642, #7700704, … |
| maximal quotient of | #576952, #576960, #577076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39872);