Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #3391 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
169 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 169 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) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,21,25)(19,22,28,31)(20,29,27,24)(23,32,30,26);(4,13)(7,12)(9,16)(11,14)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)Polycyclic presentation:
2^8|a^2=gh,c^2=g,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=eg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12384, #489214, #489215, … |
| maximal normal subgroup of | C2×#12384, #489214, #489215, … |
| maximal quotient of | #125964, #125998, #126034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5372);