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 | |
686 conjugacy classes of subgroups; only the 397 normal subgroups are shown.
This group has 686 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)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26,29,33)(27,31,35,30)(28,36,38,32)(34,39,37,40);(2,5)(6,8)(10,12)(13,21)(15,17)(18,24)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,29,38)(26,32,33,36)(27,34,35,37)(30,39,31,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,26)(27,31)(28,32)(29,33)(30,35)(34,40)(36,38)(37,39)Polycyclic presentation:
2^8|a^2=f,b^2=fg,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 | #53302⋊C3 |
| maximal subgroup of | #97304, #179349, #179350, … |
| maximal normal subgroup of | #97304, #179349, #179350, … |
| maximal quotient of | #606741, #606963, #607143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46290);