Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| automorphism | Aut(G) order 4096 |
| 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 | #459 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
289 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 99 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)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32);(1,2,5,7)(3,9,11,15)(4,13,12,6)(8,14,16,10)(17,19)(18,22)(20,27)(21,26)(23,31)(24,30)(25,29)(28,32);(2,7)(6,13)(8,16)(9,15)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(1,3,5,11)(2,8,7,16)(4,14,12,10)(6,15,13,9)(18,24)(22,30)(23,25)(29,31)Polycyclic presentation:
2^8|b^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,c^b=cgh,d^b=de,e^b=eg,f^c=fh,e^d=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20296, #6361239, #6361264, … |
| maximal normal subgroup of | C2×#20296, #6361239, #6361264, … |
| maximal quotient of | #220917, #220925, #220928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13284);