Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#106 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 207 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)(5,6,8,12,10,15,22,14)(7,16,19,33,20,34,17,31)(9,23,21,13,25,30,35,32)(11,26,28,18,29,24,27,36);(5,7,10,20)(6,11)(8,17,22,19)(9,24,25,26)(12,27)(13,31)(14,28)(15,29)(16,23)(18,35,36,21)(30,34)(32,33);(2,4)(5,8,10,22)(6,12,15,14)(7,17,20,19)(9,21,25,35)(11,27,29,28)(13,30,32,23)(16,31,34,33)(18,26,36,24)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=h,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #144718, #144719, #144720, … |
| maximal normal subgroup of | #144718, #144719, #144720, … |
| maximal quotient of | #135827, #135845, #136010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5918);