Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C24 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
293 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 159 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)(7,18)(8,14)(9,25)(10,15)(11,28)(12,17)(13,20)(16,27)(19,30)(21,29)(22,36)(23,32)(24,33)(26,31)(34,35);(5,7,10,21,11,22,27,20,12,23,24,35,8,19,9,26)(6,13,16,30,14,29,15,32,17,31,25,36,28,34,33,18);(5,8,12,11)(6,14,17,28)(7,19,23,22)(9,24,27,10)(13,29,31,34)(15,25,33,16)(18,30,32,36)(20,21,26,35)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=gh,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fg,g^a=gh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146343, #146346, #146432, … |
| maximal normal subgroup of | #146343, #146346, #146432, … |
| maximal quotient of | #126664, #126696, #126715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5555);