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:
(2,6)(3,9)(8,13)(10,15)(17,18)(19,25)(20,22)(21,28)(23,27)(24,29)(26,30)(31,32)(33,34,39,37)(35,42,36,40)(38,45,46,41)(43,47,44,48);(2,6)(7,12)(8,13)(14,16)(18,20)(21,29)(23,25)(26,32)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,37,39,34)(35,40,36,42)(38,45,46,41)(43,47,44,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)Polycyclic presentation:
2^8|a^2=fgh,b^2=fgh,c^2=f,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×#56010, #8102830, #8103198, … |
| maximal normal subgroup of | C2×#56010, #8102830, #8103198, … |
| maximal quotient of | #585785, #586644, #587161, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48998);