Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
217 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 217 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2,4,10,6,11,20,28)(3,12,15,27,16,9,18,26)(5,8,19,24,14,25,17,30)(7,21,22,29,23,31,32,13);(2,7,11,23)(3,13)(4,17)(8,12,25,9)(10,27,28,26)(15,18)(16,29)(19,20)(21,31)(22,30,32,24);(2,8)(7,12)(9,23)(10,24)(11,25)(22,27)(26,32)(28,30)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=gh,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,d^b=dg,e^b=efgh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #132950, #132951, #132953, … |
| maximal normal subgroup of | #132950, #132951, #132953, … |
| maximal quotient of | #124528, #124550, #124607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4802);