Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,22)(13,17)(14,28)(15,19)(18,29)(20,30)(21,24)(23,25)(26,38)(27,33)(31,35)(32,39)(34,36)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,22)(10,16,19,12)(13,25,27,20)(14,21,29,35)(17,23,33,30)(18,31,28,24)(26,37,39,34)(32,36,38,40);(9,12)(10,11)(13,23)(14,24)(15,16)(17,20)(18,21)(19,22)(25,33)(26,36)(27,30)(28,35)(29,31)(32,34)(37,38)(39,40);(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,30)(21,31)(23,25)(24,35)(26,32)(27,33)(28,29)(34,40)(36,37)(38,39);(9,13,15,27)(10,17,19,33)(11,20,22,25)(12,23,16,30)(14,26,29,39)(18,32,28,38)(21,34,35,37)(24,36,31,40);(1,3,4,7)(2,5,6,8)(9,14)(10,18)(11,21)(12,24)(13,26)(15,29)(16,31)(17,32)(19,28)(20,34)(22,35)(23,36)(25,37)(27,39)(30,40)(33,38)Polycyclic presentation:
2^8|a^2=g,b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #62861⋊C3 |
| maximal subgroup of | #6453890, #6453892, #6453894, … |
| maximal normal subgroup of | #6453890, #6453892, #6453894, … |
| maximal quotient of | #7625699, #7625703, #7625724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55849);