Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #376 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 135 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)(7,16,19,30)(8,13)(9,14)(10,24)(11,15)(12,18,26,20)(17,29,32,36)(21,27)(22,34)(23,28)(25,31,35,33);(5,7,10,20,11,19,9,18)(6,12,14,16,15,26,24,30)(8,17,22,33,23,32,21,31)(13,25,27,29,28,35,34,36);(2,4)(5,8,9,21,11,23,10,22)(6,13,14,27,15,28,24,34)(7,17,18,31,19,32,20,33)(12,25,16,29,26,35,30,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,g^a=gh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129909, #129910, #129911, … |
| maximal normal subgroup of | #129909, #129910, #129911, … |
| maximal quotient of | #108392, #108393, #108394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3788);