Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| 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 | |
785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29);(2,5)(3,7)(9,10,12,14)(11,15,16,13)(18,23)(19,26)(25,28)(27,32);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29);(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,13,14,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=g,d^2=f,e^2=h,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 | #130576, #130894, #254189, … |
| maximal normal subgroup of | #130576, #130894, #254189, … |
| maximal quotient of | #521627, #522261, #522916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32174);