Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10018 |
| automorphism | Aut(G) = C2×#72278229 (order 4096) · Inn(G) = #392 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #2631 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
288 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 288 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12)(3,7,14,9)(4,8,17,26)(6,13,19,11)(10,20,27,18)(15,24,28,23)(16,31,29,25)(21,32,22,30);(2,7)(3,14)(4,17)(8,25)(9,12)(10,21)(11,23)(13,24)(15,28)(18,20)(22,27)(26,31);(2,8)(3,15)(4,18)(7,21)(9,22)(10,11)(12,26)(13,27)(14,28)(16,29)(17,20)(23,25)(24,31)(30,32)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=egh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10022, #235431, #235432, … |
| maximal normal subgroup of | C2×#10022, #235431, #235432, … |
| maximal quotient of | #103006, #103009, #103012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3010);