Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | Aut(G) = #72317053 (order 4096) · Inn(G) = #486 · Out(G) = #392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #486 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2071 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
148 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(2,7,9,16)(5,19)(8,23,24,22)(10,26,25,20)(11,27,28,29)(13,30)(17,32)(21,31);(1,2)(3,11)(4,16)(5,20)(6,9)(7,18)(8,13)(10,17)(12,29)(14,27)(15,28)(19,26)(21,23)(22,31)(24,30)(25,32);(1,3,4,12,6,15,18,14)(2,8,7,22,9,24,16,23)(5,21,17,30,19,31,32,13)(10,27,20,28,25,29,26,11)Polycyclic presentation:
2^8|a^2=e,c^2=d,d^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fg,g^b=gh,f^c=fgh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #126295, #126296, #126387, … |
| maximal normal subgroup of | #126295, #126296, #126387, … |
| maximal quotient of | #146008, #146009, #146021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6474);