Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
445 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 445 conjugacy classes of subgroups — too many to draw. Only its 239 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 — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,9,24)(7,12,18,26,19,27,17,33)(8,20,23,28,21,34,22,13)(16,29,32,35,30,36,31,25);(2,4)(5,7,10,18,11,19,9,17)(6,12,14,26,15,27,24,33)(8,16,23,32,21,30,22,31)(13,25,20,29,28,35,34,36);(5,8,9,22,11,21,10,23)(6,13,14,20,15,28,24,34)(7,16,17,31,19,30,18,32)(12,25,26,29,27,35,33,36);(8,21)(13,28)(16,30)(20,34)(22,23)(25,35)(29,36)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124847, #140691, #370108, … |
| maximal normal subgroup of | #124847, #140691, #370108, … |
| maximal quotient of | #203506, #203518, #203530, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14152);