Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| 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 | #481 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 248 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:
(2,4)(5,6,9,15,26,28,23,14,11,13,24,32,10,16,8,22)(7,17,20,12,27,30,18,34,19,35,36,29,21,33,25,31);(1,2,3,4)(5,7,10,27,11,19,26,21)(6,12,28,31,13,29,16,34)(8,20,24,18,23,36,9,25)(14,33,32,35,22,30,15,17);(6,13)(8,23)(9,24)(12,29)(16,28)(18,25)(20,36)(31,34)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #131316, #131317, #131319, … |
| maximal normal subgroup of | #131316, #131317, #131319, … |
| maximal quotient of | #136617, #136645, #136890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5997);