Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| 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 39 normal subgroups are shown.
This group has 109 conjugacy classes of subgroups — too many to draw. Only its 39 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,18,19,23)(12,21,22,24)(14,20,15,17);(2,5)(6,7)(9,11,18,16,14,21,12,20,13,19,23,10,15,24,22,17);(2,6)(4,8)(5,7)(10,16)(11,19)(17,20)(21,24)Polycyclic presentation:
2^8|a^2=h,b^2=dfgh,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,c^b=ch,e^c=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12474, #490571, #490573, … |
| maximal normal subgroup of | C2×#12474, #490571, #490573, … |
| maximal quotient of | #135000, #138578, #138645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5462);