Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 75 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)(5,6,10,15,11,16,23,33,12,17,9,25,8,14,24,31)(7,18,21,29,19,32,27,30,22,35,20,34,28,36,26,13);(5,7)(6,13,17,30)(8,19)(9,26)(10,27)(11,28)(12,22)(14,29,16,34)(15,32,25,36)(18,31,35,33)(20,24)(21,23);(2,4)(5,8,12,11)(6,14,17,16)(7,19,22,28)(9,23,10,24)(13,29,30,34)(15,31,25,33)(18,32,35,36)(20,26,21,27)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=h,d^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,f^b=fg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131332, #131333, #131334, … |
| maximal normal subgroup of | #131332, #131333, #131334, … |
| maximal quotient of | #136612, #136614, #136640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5990);