Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10611 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #396 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #396 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #126 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
138 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,10,6,11,5,15)(3,12,14,16,4,9,13,8)(17,18)(19,21)(20,28)(22,25)(23,26)(24,30)(27,31)(29,32);(1,3,7,14,6,4,5,13)(2,8,15,9,11,16,10,12)(17,19,22,26,23,25,21,18)(20,27,30,24,31,28,29,32);(1,4,6,3)(2,9,11,12)(5,13,7,14)(8,10,16,15)(17,20,23,31)(18,24,26,32)(19,27,25,28)(21,30,22,29)Polycyclic presentation:
2^8|a^2=de,b^2=dg,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fh,d^c=dg,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10614, #242363, #242364, … |
| maximal normal subgroup of | C2×#10614, #242363, #242364, … |
| maximal quotient of | #105188, #105190, #105192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3602);