Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,15)(11,14,16,12)(18,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22);(2,6)(3,7)(9,11,13,16)(10,12,15,14);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,20)(18,23,22,24);(1,4)(2,7)(3,6)(5,8)(9,12,13,14)(10,11,15,16)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6446481, #6446483, #6446496, … |
| maximal normal subgroup of | #6446481, #6446483, #6446496, … |
| maximal quotient of | #515404, #515413, #515489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28568);