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, cosocle | C23 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,12)(13,15)(14,16);(1,3,4,8)(2,5,6,7)(9,11,13,16)(10,12,15,14)(19,23)(21,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,18,20,22)(19,21,23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24);(1,2)(3,5)(4,6)(7,8)(9,12,13,14)(10,11,15,16)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=gh,e^2=fg,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 | #7007270, #7007282, #7007323, … |
| maximal normal subgroup of | #7007270, #7007282, #7007323, … |
| maximal quotient of | #531790, #531941, #532147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52555);