Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| 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 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 69 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)(5,6,10,14,26,35,9,27,12,18,8,15,11,17,28,16)(7,19,22,33,29,30,25,13,24,32,20,36,23,31,21,34);(1,2,3,4)(5,7)(6,13)(8,25)(9,20)(10,21)(11,29)(12,24)(14,30)(15,31)(16,32)(17,36)(18,34)(19,27)(22,28)(23,26)(33,35);(6,14)(8,9)(10,28)(11,26)(13,30)(15,18)(16,35)(17,27)(19,36)(20,25)(21,22)(23,29)(31,34)(32,33)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=egh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13019, #494517, #494518, … |
| maximal normal subgroup of | C2×#13019, #494517, #494518, … |
| maximal quotient of | #136691, #136725, #136900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6007);