Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,4,9,5,10,15,23)(3,8,12,21,11,22,24,32)(6,7,14,18,16,19,27,29)(13,17,26,28,25,20,30,31);(2,7)(3,11)(8,20)(10,19)(12,24)(13,25)(17,22)(21,32)(26,30)(28,31);(1,3,6,13)(2,8,7,17)(4,12,14,26)(5,11,16,25)(9,21,18,28)(10,22,19,20)(15,24,27,30)(23,32,29,31)Polycyclic presentation:
2^8|a^2=e,c^2=h,e^2=g,b^a=bd,d^a=df,f^a=fh,c^b=cg,e^b=ef,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132821, #132823, #132825, … |
| maximal normal subgroup of | #132821, #132823, #132825, … |
| maximal quotient of | #125069, #125087, #125103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4530);