Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ 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:
(2,3)(5,6)(7,11)(8,18,19,12)(9,14)(10,13)(15,27,28,22)(16,24)(17,23)(20,31,32,25)(21,26)(29,35,36,33)(30,34);(1,2,4,3)(5,7,9,16,10,17,21,30)(6,11,14,24,13,23,26,34)(8,15,20,29,19,28,32,36)(12,22,25,33,18,27,31,35);(1,3,4,2)(5,8)(6,12)(7,15)(9,20)(10,19)(11,22)(13,18)(14,25)(16,29)(17,28)(21,32)(23,27)(24,33)(26,31)(30,36)(34,35)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93369, #128802, #128803, … |
| maximal normal subgroup of | #93369, #128802, #128803, … |
| maximal quotient of | #108400, #108401, #108402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3791);