Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13545 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2101 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2101 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 47 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:
(1,2,5,10,6,8,18,25)(3,11,21,24,13,7,15,28)(4,16,19,32,20,9,17,27)(12,26,14,23,30,22,31,29);(1,3)(2,7,8,11)(4,12)(5,21)(6,13)(9,26,16,22)(10,28,25,24)(14,19)(15,18)(17,31)(20,30)(23,32,29,27);(2,8)(3,12,13,30)(4,17)(5,18)(7,22,11,26)(9,27)(14,21,31,15)(16,32)(19,20)(23,24,29,28)Polycyclic presentation:
2^8|a^2=dg,b^2=de,c^2=d,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,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 | C2×#13545, #499562, #499603, … |
| maximal normal subgroup of | C2×#13545, #499562, #499603, … |
| maximal quotient of | #146288, #146289, #148746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6533);