Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13340 |
| automorphism | Aut(G) = #72317053 (order 4096) · Inn(G) = #493 · Out(G) = #392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #493 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2093 |
| 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 | |
156 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(1,2,5,11,16,26,6,12,7,14,22,32,8,13,24,31)(3,15,17,30,25,29,23,27,19,28,4,21,20,10,18,9);(1,3,16,20,7,19,8,25)(2,9,26,21,14,27,13,30)(4,22,23,24,17,5,18,6)(10,12,28,32,29,31,15,11);(1,4)(2,10)(3,5)(6,20)(7,17)(8,18)(9,12)(11,30)(13,15)(14,29)(16,23)(19,22)(21,32)(24,25)(26,28)(27,31)Polycyclic presentation:
2^8|a^2=d,b^2=deg,d^2=gh,e^2=g,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dgh,e^c=egh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #126295, #126296, #126437, … |
| maximal normal subgroup of | #126295, #126296, #126437, … |
| maximal quotient of | #146190, #146191, #146192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6520);