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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,7,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32)(33,34)(35,37)(36,43)(38,40)(39,41)(42,45)(44,47)(46,48);(3,10)(6,9)(11,16)(13,15)(18,23)(20,24)(26,29)(28,31)(33,35,39,36)(34,37,41,43)(38,44,46,42)(40,47,48,45);(1,3)(2,6)(4,11)(5,10)(7,13)(8,9)(12,16)(14,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,36,39,35)(34,37,41,43)(38,44,46,42)(40,45,48,47);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,48)(46,47);(1,4,5,12)(2,7,8,14)(3,11,10,16)(6,13,9,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,38,39,46)(34,40,41,48)(35,42,36,44)(37,45,43,47)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=fgh,d^2=f,e^2=g,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×#57818, #8101754, #8101965, … |
| maximal normal subgroup of | C2×#57818, #8101754, #8101965, … |
| maximal quotient of | #586334, #587166, #587256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50806);