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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,9)(8,13)(11,15)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(34,38)(35,41)(36,44)(46,48);(4,12)(8,13)(11,15)(14,16)(18,20)(23,25)(24,28)(30,31)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=fh,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×#57754, #8101542, #8101961, … |
| maximal normal subgroup of | C2×#57754, #8101542, #8101961, … |
| maximal quotient of | #586772, #586920, #587137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50742);