Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12117 |
| automorphism | Aut(G) = #72308067 (order 2048) · Inn(G) = #1595 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1595 |
| 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 | |
212 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 212 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)(3,14)(4,8)(5,12)(6,11)(7,13)(9,16)(10,18)(15,27)(17,28)(19,24)(20,26)(21,23)(22,31)(25,32)(29,30);(2,8)(4,18)(5,6)(9,22)(10,12)(11,23)(13,24)(14,27)(15,30)(16,20)(19,21)(25,28)(26,29)(31,32);(1,3)(2,9)(4,15)(5,20)(6,16)(7,17)(8,22)(10,27)(11,28)(12,14)(13,26)(18,30)(19,32)(21,31)(23,25)(24,29)Polycyclic presentation:
2^8|d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,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 | #151150, #151151, #151154, … |
| maximal normal subgroup of | #151150, #151151, #151154, … |
| maximal quotient of | #123011, #123015, #123019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5105);