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:
(1,2,6,4)(3,7,11,9)(5,12,13,8)(10,15,16,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(2,4)(7,9)(8,12)(14,15)(21,29)(24,31)(26,32)(28,30)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^8|a^2=fgh,b^2=h,c^2=fgh,d^2=fg,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×#57937, #8101434, #8101956, … |
| maximal normal subgroup of | C2×#57937, #8101434, #8101956, … |
| maximal quotient of | #586328, #586534, #587134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50925);