Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,12,14)(11,13,15,16)(18,20)(19,25)(24,28)(27,31);(2,5)(3,7)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32);(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6548236, #6548238, #6548250, … |
| maximal normal subgroup of | #6548236, #6548238, #6548250, … |
| maximal quotient of | #521687, #522309, #522819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31203);