Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
873 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 873 conjugacy classes of subgroups — too many to draw. Only its 447 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) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10,13,15)(11,12,16,14)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(18,19)(23,25)(24,26)(30,31);(2,5)(6,8)(9,11,13,16)(10,14,15,12)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fg,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #122109, #122143, #122193, … |
| maximal normal subgroup of | #122109, #122143, #122193, … |
| maximal quotient of | #556413, #556422, #556623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30469);