Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator | C24 | C22×C4 |
| Frattini, prime radical, cosocle | C25 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 425 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:
(5,6)(7,13,15,10)(8,16)(9,12)(11,18)(14,19,17,20);(1,2)(3,4)(5,7)(6,10)(8,17,18,14)(9,15)(11,20,16,19)(12,13);(2,4)(5,8)(6,11,12,16)(7,14)(9,18)(10,19,13,20)(15,17)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11279, #358293, C4×#2032, … |
| maximal normal subgroup of | C2×#11279, #358293, C4×#2032, … |
| maximal quotient of | #120633, #120634, #120727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4267);