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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(2,5)(6,8)(13,21)(16,20)(18,24)(22,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40)Polycyclic presentation:
2^8|a^2=g,b^2=f,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 | #53307⋊C3 |
| maximal subgroup of | C2×#53307, #8105794, #8105940, … |
| maximal normal subgroup of | C2×#53307, #8105794, #8105940, … |
| maximal quotient of | #606724, #606859, #606902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46295);