Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12298 |
| automorphism | Aut(G) = C2×#6346637 (order 1024) · Inn(G) = #1613 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1613 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #3197 | 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 | |
164 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 164 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:
(1,2,6,21,7,11,4,9)(3,12,13,26,15,30,22,29)(5,17,20,10,19,27,16,28)(8,18,23,14,25,31,24,32);(1,3)(2,8)(4,13)(5,18)(6,22)(7,15)(9,23)(10,26)(11,25)(12,27)(14,16)(17,30)(19,31)(20,32)(21,24)(28,29);(1,4,7,6)(2,9,11,21)(3,13,15,22)(5,16,19,20)(8,23,25,24)(10,27,28,17)(12,29,30,26)(14,31,32,18)Polycyclic presentation:
2^8|a^2=fg,c^2=h,d^2=f,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12300, #484569, #484570, … |
| maximal normal subgroup of | C2×#12300, #484569, #484570, … |
| maximal quotient of | #126284, #126440, #126441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5288);