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 | |
633 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,9)(8,13)(11,15)(17,18)(19,25)(20,27)(21,29)(22,23)(24,30)(26,28)(31,32)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,48,46,44);(4,12)(8,13)(11,15)(14,16)(18,23)(20,25)(21,30)(26,32)(33,35,38,40)(34,39,42,36)(37,46,47,43)(41,44,45,48);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,29)(25,31)(27,32)(33,37,38,47)(34,41,42,45)(35,43,40,46)(36,44,39,48)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,d^2=fgh,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×#58557, #8104264, #8104426, … |
| maximal normal subgroup of | C2×#58557, #8104264, #8104426, … |
| maximal quotient of | #586365, #586589, #586908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51545);