Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #333 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | C2×#333 |
| prime radical | C23 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,5,9,18,10,28,26,6,11,20,27,4,8,17,24)(3,12,15,25,13,23,19,31,14,29,30,32,16,7,21,22);(1,3,6,14)(2,7)(4,13,18,16)(5,19,20,21)(8,12)(9,25)(10,29)(11,23)(15,17,30,28)(22,26)(24,31)(27,32);(1,4,6,18)(2,8,11,10)(3,13,14,16)(5,17,20,28)(7,12,23,29)(9,24,27,26)(15,19,30,21)(22,25,31,32)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=h,d^2=gh,e^2=fg,f^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efg,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #128907, #128925, #128947, … |
| maximal normal subgroup of | #128907, #128925, #128947, … |
| maximal quotient of | #124129, #124241, #124984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4944);