Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1893 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1893 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,6,11,8,4,7,19)(3,12,16,28,17,29,15,27)(5,20,21,18,23,32,22,10)(9,24,26,31,13,30,25,14);(1,3,5,14,6,16,21,24,8,17,23,31,7,15,22,30)(2,9,18,29,19,25,20,28,4,13,10,12,11,26,32,27);(1,4)(2,8)(3,13)(5,10)(6,11)(7,19)(9,17)(12,14)(15,25)(16,26)(18,23)(20,22)(21,32)(24,27)(28,30)(29,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^b=eh,d^c=dh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13070, #494653, #495899, … |
| maximal normal subgroup of | C2×#13070, #494653, #495899, … |
| maximal quotient of | #137615, #137616, #137655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6058);