Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #400 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(2,3)(5,6)(7,12,8,18)(9,16)(10,14)(11,15)(13,19,20,17);(1,2,4,3)(5,7,10,17,9,8,11,19)(6,12,14,13,16,18,15,20);(1,3,4,2)(5,8,9,7)(6,13)(10,19,11,17)(12,15)(14,18)(16,20)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11261, #358272, #358420, … |
| maximal normal subgroup of | C2×#11261, #358272, #358420, … |
| maximal quotient of | #120683, #120684, #120703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4249);