Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(12,19)(16,21)(18,23)(22,24)(26,30)(27,33)(32,37)(34,39);(10,14)(11,17)(16,21)(18,23)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^8|b^2=f,c^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37177, #7634480, #7634501, … |
| maximal normal subgroup of | C2×#37177, #7634480, #7634501, … |
| maximal quotient of | #553943, #554136, #554137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30165);