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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
643 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)(17,18,21,22)(19,23,25,29)(20,24,27,30)(26,31,28,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(4,12)(8,13)(11,15)(14,16)(18,22)(19,25)(20,27)(31,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19)(18,23)(20,28)(21,25)(22,29)(24,32)(26,27)(30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,24)(19,26)(21,27)(22,30)(23,31)(25,28)(29,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=fgh,d^2=g,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×#57906, #8101754, #8101955, … |
| maximal normal subgroup of | C2×#57906, #8101754, #8101955, … |
| maximal quotient of | #586400, #586462, #587134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50894);