Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
583 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 583 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,9,15,11,16,10,14)(7,17,21,26,20,27,19,12)(8,13,23,28,24,30,22,29)(18,32,33,25,34,31,35,36);(5,7,10,19,11,20,9,21)(6,12,14,27,16,26,15,17)(8,18,22,35,24,34,23,33)(13,25,29,32,30,36,28,31);(5,8)(6,13)(7,18)(9,23)(10,22)(11,24)(12,25)(14,29)(15,28)(16,30)(17,31)(19,35)(20,34)(21,33)(26,36)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94093, #94095, #94097, … |
| maximal normal subgroup of | #94093, #94095, #94097, … |
| maximal quotient of | #105772, #105773, #105776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3741);