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 | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,12,19,16)(13,21,22,17)(18,23,24,20)(25,26,30,33)(27,28,31,36)(29,32,38,40)(34,35,37,39);(2,4)(3,7)(10,15)(12,16)(13,22)(18,24)(25,27,30,31)(26,28,33,36)(29,37,38,34)(32,39,40,35);(1,3,5,7)(2,6,4,8)(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,27)(26,28)(29,34)(30,31)(32,35)(33,36)(37,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,37)(34,40)(38,39);(1,4,5,2)(3,8,7,6)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,40)(36,39)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=f,d^2=g,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×#56182, #8102806, #8103194, … |
| maximal normal subgroup of | C2×#56182, #8102806, #8103194, … |
| maximal quotient of | #585938, #586481, #586819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49170);