Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(1,2,4,6)(3,5,7,8)(10,14)(12,21)(15,17)(18,24)(26,28)(31,33)(32,36)(38,39);(1,3)(2,5)(4,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50197, #7706143, #7706194, … |
| maximal normal subgroup of | C2×#50197, #7706143, #7706194, … |
| maximal quotient of | #560419, #560589, #561427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43185);