Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
807 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)(9,10,13,14)(11,16,15,12);(2,5)(6,7)(10,14)(11,15)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,14)(13,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6687901, #6687907, #6687945, … |
| maximal normal subgroup of | #6687901, #6687907, #6687945, … |
| maximal quotient of | #544560, #545104, #545560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35804);