Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) order 65536 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26,30,34)(27,32,28,31)(29,37,38,33)(35,39,36,40);(1,2)(3,7)(4,8)(5,6)(10,14)(11,18)(16,20)(19,24)(25,27,30,28)(26,31,34,32)(29,36,38,35)(33,40,37,39);(2,6)(3,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,26)(27,32)(28,31)(29,33)(30,34)(35,40)(36,39)(37,38);(1,4)(2,3)(5,7)(6,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=fh,d^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 | #133056, #487168, #487296, … |
| maximal normal subgroup of | #133056, #487168, #487296, … |
| maximal quotient of | #585435, #586957, #587396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51779);