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 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,29)(28,36)(30,33)(31,35)(32,38)(34,39)(37,40);(2,5)(3,8)(12,20)(15,22)(18,24)(21,23)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(2,5)(6,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,3)(2,6)(4,8)(5,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=fgh,d^2=h,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×#58833, #8105074, #8105136, … |
| maximal normal subgroup of | C2×#58833, #8105074, #8105136, … |
| maximal quotient of | #585413, #586272, #586357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51821);