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 | #2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
727 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,2)(3,7)(4,6)(5,8)(11,18)(14,17)(19,24)(21,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=fgh,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×#56553, #8099012, #8099314, … |
| maximal normal subgroup of | C2×#56553, #8099012, #8099314, … |
| maximal quotient of | #587141, #587152, #587287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49541);