Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12170 |
| automorphism | Aut(G) = #72308439 (order 2048) · Inn(G) = #1598 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1598 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
178 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 35 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)(3,12)(4,7)(5,20)(6,11)(8,14)(9,16)(10,18)(13,26)(15,25)(17,21)(19,22)(23,32)(24,31)(27,29)(28,30);(2,7)(4,16)(5,18)(8,21)(9,10)(11,22)(12,26)(13,29)(15,30)(17,31)(19,32)(20,23)(24,25)(27,28);(1,3,6,14)(2,8)(4,17,19,13)(5,15,18,30)(7,21)(9,24)(10,25)(11,12)(16,31,32,29)(20,28)(22,26)(23,27)Polycyclic presentation:
2^8|c^2=g,d^2=fh,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gh,d^b=df,e^b=egh,f^b=fh,d^c=dg,e^c=eh,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12170, #484651, #484652, … |
| maximal normal subgroup of | C2×#12170, #484651, #484652, … |
| maximal quotient of | #123171, #123175, #123179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5158);