Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C4⋊C8 |
| prime radical | C24 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
293 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 175 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)(5,6,9,14,10,13,23,28,11,15,24,29,8,21,22,34)(7,16,19,31,20,26,33,35,17,30,32,36,18,12,25,27);(2,4)(5,7)(6,12,15,26)(8,18)(9,19)(10,20)(11,17)(13,16,21,30)(14,27,29,35)(22,25)(23,33)(24,32)(28,31,34,36);(7,17)(12,26)(16,30)(18,20)(19,32)(25,33)(27,35)(31,36)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=dh,f^a=fh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #124350, #124351, #125381, … |
| maximal normal subgroup of | #124350, #124351, #125381, … |
| maximal quotient of | #126553, #126565, #126579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5537);