Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
163 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(5,6,11,28,12,18,26,17)(7,19,24,31,25,35,36,33)(8,14,20,15,9,16,10,27)(13,29,34,21,30,23,32,22);(1,2)(3,4)(5,7,20,29,11,24,9,21,12,25,10,23,26,36,8,22)(6,13,14,33,17,32,27,35,18,30,16,31,28,34,15,19);(2,4)(6,14)(7,21)(8,20)(9,10)(11,26)(13,30)(15,17)(16,18)(19,31)(22,25)(23,36)(24,29)(27,28)(33,35)Polycyclic presentation:
2^8|a^2=de,b^2=dg,d^2=gh,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,d^c=dg,e^c=eg,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13470, #499313, #499314, … |
| maximal normal subgroup of | C2×#13470, #499313, #499314, … |
| maximal quotient of | #145904, #145914, #145924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6458);