Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,8,7,5)(9,10,13,11)(12,14,16,15)(17,18,22,19)(20,24,21,23);(1,3)(2,5)(4,8)(6,7)(12,16)(14,15)(17,19,22,18)(20,23,21,24);(9,11,13,10)(12,15,16,14)(17,20,22,21)(18,23,19,24);(1,4,6,2)(3,8,7,5)(9,12)(10,14)(11,15)(13,16)(17,21,22,20)(18,24,19,23);(1,5,6,8)(2,7,4,3)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #7117961, #7117967, #7117977, … |
| maximal normal subgroup of | #7117961, #7117967, #7117977, … |
| maximal quotient of | #531896, #531945, #532058, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52659);