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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
196 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 196 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,12,18)(7,19,22,31)(8,23,20,13)(9,17,11,15)(10,26,29,35)(14,32,34,24)(16,25,36,28)(21,33,27,30);(1,2)(3,4)(5,7)(6,13)(8,9)(10,27)(11,20)(12,22)(14,26)(15,31)(16,33)(17,19)(18,23)(21,29)(24,28)(25,32)(30,36)(34,35);(2,4)(5,8,9,7,12,20,11,22)(6,14,15,30,18,34,17,33)(10,27,25,24,29,21,28,32)(13,16,19,35,23,36,31,26)Polycyclic presentation:
2^8|a^2=h,c^2=d,d^2=h,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,e^c=egh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13394, #499032, #499034, … |
| maximal normal subgroup of | C2×#13394, #499032, #499034, … |
| maximal quotient of | #144689, #144781, #145003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6382);