Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 196608 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
682 conjugacy classes of subgroups; only the 397 normal subgroups are shown.
This group has 682 conjugacy classes of subgroups — too many to draw. Only its 397 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(2,5)(6,8)(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)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,16,17)(12,19,20,24)(15,21,22,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,2)(3,6)(4,5)(7,8)(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=g,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #53300⋊C3 |
| maximal subgroup of | #97306, #179360, #179361, … |
| maximal normal subgroup of | #97306, #179360, #179361, … |
| maximal quotient of | #606741, #606930, #607168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46288);