Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13667 |
| automorphism | Aut(G) = #72285396 (order 2048) · Inn(G) = #2184 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
147 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 147 conjugacy classes of subgroups — too many to draw. Only its 37 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:
(2,7)(3,11)(4,14)(5,17)(8,20)(9,23)(10,24)(12,16)(13,25)(15,30)(18,29)(19,21)(22,31)(26,27)(28,32);(1,2)(3,12)(4,16)(5,19)(6,8)(7,11)(9,14)(10,17)(13,28)(15,31)(18,32)(20,25)(21,30)(22,29)(23,26)(24,27);(3,13)(7,20)(11,25)(12,28)(15,26)(21,24)(23,31)(27,30)Polycyclic presentation:
2^8|d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,d^b=df,e^b=eh,f^b=fg,g^b=gh,d^c=dh31 conjugacy classes — view the full character table.
| maximal subgroup of | #151182, #151184, #151185, … |
| maximal normal subgroup of | #151182, #151184, #151185, … |
| maximal quotient of | #151170, #151171, #151172, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6655);