Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(10,14)(11,17)(16,21)(18,23)(26,30)(27,33)(28,35)(38,40);(10,14)(11,17)(12,19)(22,24)(25,26,29,30)(27,31,33,37)(28,36,35,32)(34,40,39,38);(1,3,5,8)(2,6,4,7)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,33)(26,31,30,37)(28,34,35,39)(32,38,36,40);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,35)(26,32,30,36)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=fg,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46185, #7702836, #7702845, … |
| maximal normal subgroup of | C2×#46185, #7702836, #7702845, … |
| maximal quotient of | #557281, #557332, #557505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39173);