Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | |
696 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)(6,8)(10,14)(12,20)(15,17)(18,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,2,5,4)(3,6,7,8)(10,14)(15,17)(16,21)(22,23)(26,31)(27,34)(33,36)(35,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,4,5,2)(3,8,7,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37577, #7633825, #7633828, … |
| maximal normal subgroup of | C2×#37577, #7633825, #7633828, … |
| maximal quotient of | #556456, #556459, #556461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30565);