Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | #432 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 247 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) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,11,13,12)(14,18,20,19);(2,4)(5,6,8,14,9,11,17,18,10,13,16,20,7,12,15,19);(6,12,13,11)(8,16)(14,18,20,19)(15,17)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=h,f^2=g,g^2=h,b^a=bd,c^b=cg,d^b=dh,e^b=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93545, #93574, #124832, … |
| maximal normal subgroup of | #93545, #93574, #124832, … |
| maximal quotient of | #128261, #128279, #128354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5595);