Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3150 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
279 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(5,6,13,20)(7,15,25,35)(8,26,21,14)(9,17,29,36)(10,16,27,30)(11,19,12,18)(22,33,28,32)(23,34,24,31);(1,2)(3,4)(5,7)(6,14)(8,27)(9,28)(10,21)(11,24)(12,23)(13,25)(15,17)(16,34)(18,33)(19,32)(20,26)(22,29)(30,31)(35,36);(2,4)(5,8)(6,15)(7,9)(10,23)(11,22)(12,28)(13,21)(14,30)(16,26)(17,33)(18,34)(19,31)(20,35)(24,27)(25,29)(32,36)Polycyclic presentation:
2^8|a^2=h,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dgh,e^c=egh,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125645, #125646, #125675, … |
| maximal normal subgroup of | #125645, #125646, #125675, … |
| maximal quotient of | #142642, #142643, #142722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6330);