Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,2,4,6)(3,5,7,8)(10,14)(12,20)(15,17)(18,24)(26,31)(27,32)(33,38)(34,39);(1,3)(2,5)(4,7)(6,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)Polycyclic presentation:
2^8|b^2=g,c^2=h,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#42324, #7640811, #7640827, … |
| maximal normal subgroup of | C2×#42324, #7640811, #7640827, … |
| maximal quotient of | #544471, #545052, #545531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35312);