Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | |
776 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)(9,10,13,16)(11,14,12,15)(17,18,22,26)(19,23,25,21)(20,24,30,32)(27,31,28,29);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(19,25)(21,23)(27,28)(29,31);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,19,22,25)(18,23,26,21)(20,28,30,27)(24,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,28,32,27)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=h,d^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7007812, #7007824, #7007869, … |
| maximal normal subgroup of | #7007812, #7007824, #7007869, … |
| maximal quotient of | #535452, #535618, #536118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38403);