Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12173 |
| automorphism | Aut(G) = C2×#6346637 (order 1024) · Inn(G) = #1598 · Out(G) = C23 |
| 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 |
| prime radical | #2190 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
173 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 173 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)(3,13)(4,17)(5,9)(6,11)(7,12)(8,14)(10,20)(15,29)(16,30)(18,23)(19,28)(21,27)(22,26)(24,32)(25,31);(1,3)(2,8)(4,18)(5,15)(6,22)(7,16)(9,23)(10,26)(11,24)(12,25)(13,17)(14,20)(19,32)(21,31)(27,29)(28,30);(2,9,12,11)(4,19)(8,23,25,24)(10,17,27,28)(13,29,30,26)(14,31)(18,32)(20,21)Polycyclic presentation:
2^8|c^2=gh,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^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12175, #484657, #484658, … |
| maximal normal subgroup of | C2×#12175, #484657, #484658, … |
| maximal quotient of | #123133, #123136, #123137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5163);