Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(5,6,8,12)(7,17)(9,23,22,33)(10,16,11,15)(13,29)(14,25,28,26)(18,34)(19,27)(20,32)(21,35)(24,31)(30,36);(1,2)(3,4)(6,12)(7,18,21,20)(13,30,27,31)(14,28)(15,16)(17,34,35,32)(19,24,29,36)(23,33);(2,4)(5,7)(6,13)(8,21)(9,24)(10,20)(11,18)(12,27)(14,32)(15,30)(16,31)(17,33)(19,25)(22,36)(23,35)(26,29)(28,34)Polycyclic presentation:
2^8|a^2=d,b^2=dh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^c=dh,e^c=egh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13372, #498965, #498966, … |
| maximal normal subgroup of | C2×#13372, #498965, #498966, … |
| maximal quotient of | #144453, #144483, #144522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6360);