Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 98304 |
| 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×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(10,14)(11,17)(16,21)(18,23)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(2,5)(6,8)(10,14)(15,20)(16,21)(22,24)(26,28)(27,34)(32,36)(35,40);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^8|c^2=fg,d^2=g,e^2=fg,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 | #53304⋊C3 |
| maximal subgroup of | C2×#53304, #8105788, #8105940, … |
| maximal normal subgroup of | C2×#53304, #8105788, #8105940, … |
| maximal quotient of | #606724, #606875, #606889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46292);