Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,7,5,8)(10,12)(11,13);(2,4)(3,5)(9,10)(11,15)(12,14)(13,16)(18,22)(20,28)(23,25)(26,32);(1,3,6,5)(2,7,4,8)(9,11,14,13)(10,15,12,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,4,6,2)(3,7,5,8)(9,12,14,10)(11,15,13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,5,6,3)(2,7,4,8)(9,13,14,11)(10,15,12,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=fg,c^2=gh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132820, #487532, #488271, … |
| maximal normal subgroup of | #132820, #487532, #488271, … |
| maximal quotient of | #585536, #585632, #587488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51579);