Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups; only the 291 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 291 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:
(5,6,10,15,11,13,9,23)(7,16,18,12,19,27,24,25)(8,14,22,29,20,28,21,33)(17,30,32,26,31,36,34,35);(1,2,4,3)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,25)(14,26)(15,27)(16,23)(20,31)(21,34)(22,32)(28,35)(29,36)(30,33);(1,3,4,2)(6,13)(8,20)(12,25)(15,23)(16,27)(17,31)(21,22)(32,34);(5,8,10,22,11,20,9,21)(6,14,15,29,13,28,23,33)(7,17,18,32,19,31,24,34)(12,26,27,36,25,35,16,30)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=h,b^a=be,c^a=ch,e^a=eh,e^b=eh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #134392, #138726, #138727, … |
| maximal normal subgroup of | #134392, #138726, #138727, … |
| maximal quotient of | #183877, #183878, #183879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13501);