Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 524288 |
| 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 | |
769 conjugacy classes of subgroups; only the 463 normal subgroups are shown.
This group has 769 conjugacy classes of subgroups — too many to draw. Only its 463 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(10,14)(11,16)(20,27)(24,29)(26,31)(30,32);(9,10,13,14)(11,15,16,12)(18,22)(19,25)(24,29)(26,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,3,4,7)(2,5,6,8)(9,12,13,15)(10,11,14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=h,d^2=g,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #122000, #122002, #122116, … |
| maximal normal subgroup of | #122000, #122002, #122116, … |
| maximal quotient of | #516534, #516703, #517086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28333);