Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12430 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | C2×D4 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
109 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 109 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7)(3,5,8,6)(10,14)(11,18)(12,19)(15,20)(16,24)(17,23)(21,22);(9,10,15,24,12,16,11,14,13,17,18,22,19,21,20,23);(2,5)(3,8)(6,7)Polycyclic presentation:
2^8|a^2=g,b^2=dfh,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,e^c=eg49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12453, #490538, #490543, … |
| maximal normal subgroup of | C2×#12453, #490538, #490543, … |
| maximal quotient of | #134960, #138576, #138577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5441);