Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | Aut(G) = #72279473 (order 2048) · Inn(G) = #1619 · Out(G) = C22⋊C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #3291 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
181 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 181 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10)(3,11,14,30)(4,16,19,26)(5,20,21,9)(7,22,24,13)(8,17,28,32)(12,29,15,27)(18,25,31,23);(4,17)(5,21)(8,26)(9,20)(11,29)(12,15)(13,31)(16,28)(18,22)(19,32)(23,25)(27,30);(1,3)(2,7)(4,18)(5,12)(6,14)(8,27)(9,23)(10,24)(11,16)(13,32)(15,21)(17,22)(19,31)(20,25)(26,30)(28,29)Polycyclic presentation:
2^8|a^2=h,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=egh,g^a=gh,d^b=df,e^b=eg,g^b=gh,d^c=dg,e^c=egh,f^c=fh,g^c=gh28 conjugacy classes — view the full character table.
| maximal subgroup of | #153626, #153627, #153629, … |
| maximal normal subgroup of | #153626, #153627, #153629, … |
| maximal quotient of | #123347, #123348, #123349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5303);