Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #493 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
295 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 101 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 — 4 generators on 32 points:
(2,6)(4,12)(7,13)(8,15)(9,16)(10,14)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(1,2,5,7)(3,9,11,15)(4,13,12,6)(8,14,16,10)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(2,7)(6,13)(8,16)(9,15)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(1,3,5,11)(2,8,7,16)(4,14,12,10)(6,15,13,9)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27)Polycyclic presentation:
2^8|b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=de,e^b=eg,f^c=fh,e^d=eg43 conjugacy classes — view the full character table.
| maximal subgroup of | #123332, #123340, #123348, … |
| maximal normal subgroup of | #123332, #123340, #123348, … |
| maximal quotient of | #220799, #220965, #225555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12953);