Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6,8,10)(7,11,12,9)(13,14,16,19)(15,23,24,41)(17,29,27,42)(18,32,28,22)(20,36,34,25)(21,38,35,31)(26,43,33,39)(30,40,44,37);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(20,34)(21,35)(22,32)(23,41)(29,42)(37,40)(39,43);(2,4)(13,15,16,24)(14,20,19,34)(17,30,27,44)(18,33,28,26)(21,39,35,43)(22,40,32,37)(23,42,41,29)(25,38,36,31)Polycyclic presentation:
2^8|a^2=d,b^2=dh,c^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^c=eg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13366, #498949, #498951, … |
| maximal normal subgroup of | C2×#13366, #498949, #498951, … |
| maximal quotient of | #144447, #144479, #144518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6354);