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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,16,15);(2,5)(6,7)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,23,20,22);(9,10)(11,13)(12,14)(15,16)(17,19,21,20)(18,22,24,23);(1,3)(2,6)(4,8)(5,7);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=gh,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 | #97779, #265702, #265706, … |
| maximal normal subgroup of | #97779, #265702, #265706, … |
| maximal quotient of | #521489, #522199, #522739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32608);