Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| 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)(7,10,14,18)(8,15)(9,12)(11,16)(13,17,19,20);(1,2)(3,4)(5,7)(6,10,12,18)(8,13)(9,14)(11,17,15,20)(16,19);(2,4)(5,8)(6,11,12,15)(7,13)(9,16)(10,17,18,20)(14,19)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^b=eh,g^b=gh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132019, #132020, #132021, … |
| maximal normal subgroup of | #132019, #132020, #132021, … |
| maximal quotient of | #121663, #121664, #121751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4339);