Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
198 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,17,12,18,11,29)(7,19,22,35,24,33,23,30)(8,16,26,34,27,13,9,15)(14,28,31,21,32,20,36,25);(2,4)(6,13,18,16)(8,9)(10,11)(14,30,32,35)(15,17,34,29)(19,31,33,36)(20,21)(22,23)(25,28)(26,27);(5,7,12,24)(6,14,18,32)(8,25,27,21)(9,28,26,20)(10,23,11,22)(13,30,16,35)(15,33,34,19)(17,36,29,31)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,d^b=dg,e^b=egh,f^b=fh,g^b=gh,d^c=dgh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12957, #494339, #494341, … |
| maximal normal subgroup of | C2×#12957, #494339, #494341, … |
| maximal quotient of | #135909, #135933, #136227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5945);