Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1770 |
| prime radical | C23 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
148 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,12,21,27,5,10,18,26,20,4,9,19)(3,8,14,23,15,24,29,32,16,13,22,28,30,17,25,31);(1,3)(2,8,10,13)(4,17,12,24)(5,16)(6,14)(7,15)(9,25)(11,23,26,28)(18,22)(19,31,27,32)(20,30)(21,29);(1,4,18,27,7,2,9,26,5,12,6,19,20,10,21,11)(3,13,14,28,15,17,29,31,16,8,22,23,30,24,25,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,e^2=g,g^2=h,b^a=bd,f^a=fh,c^b=cg,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128713, #128714, #128725, … |
| maximal normal subgroup of | #128713, #128714, #128725, … |
| maximal quotient of | #129051, #129052, #129063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5655);