Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
698 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)(3,8)(4,7)(5,6)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26,31,29)(27,34,28,32)(30,37,38,33)(35,39,36,40);(1,3)(2,7)(4,6)(5,8)(11,18)(14,17)(19,24)(21,23)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(3,4)(7,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,4,6,3)(2,7,5,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(1,5,6,2)(3,7,4,8)(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=fgh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#57829, #8101488, #8101965, … |
| maximal normal subgroup of | C2×#57829, #8101488, #8101965, … |
| maximal quotient of | #585413, #586358, #587160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50817);