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 | |
789 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)(3,7)(4,5)(6,8)(17,18,23,26)(19,24,20,21)(22,25,30,32)(27,31,28,29);(2,5)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,23,20)(18,24,26,21)(22,28,30,27)(25,29,32,31);(10,13)(14,15)(17,20,23,19)(18,24,26,21)(22,28,30,27)(25,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,24)(18,20,26,19)(22,29,30,31)(25,28,32,27);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=g,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 | #132554, #487378, #487954, … |
| maximal normal subgroup of | #132554, #487378, #487954, … |
| maximal quotient of | #585535, #585882, #587475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46741);