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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,34)(35,40)(36,37)(38,41)(39,42)(43,47)(44,45)(46,48);(2,6)(3,10)(8,12)(11,16)(18,21)(20,29)(23,25)(26,30)(33,35,39,37)(34,40,42,36)(38,45,46,43)(41,44,48,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19)(18,23)(20,30)(21,25)(22,27)(24,32)(26,29)(28,31)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30)(33,37,39,35)(34,40,42,36)(38,45,46,43)(41,47,48,44);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,32,30,31)(33,38,39,46)(34,41,42,48)(35,43,37,45)(36,44,40,47)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=h,d^2=fg,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×#57055, #8100102, #8100256, … |
| maximal normal subgroup of | C2×#57055, #8100102, #8100256, … |
| maximal quotient of | #585771, #586431, #586791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50043);