Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11799 |
| 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 | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 248 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)(3,11,14,28)(4,16,19,9)(5,8)(6,10)(7,13,22,17)(12,29,15,26)(18,25)(20,27,32,24)(21,31,23,30);(1,3)(2,7,10,23)(4,17)(5,15)(6,12)(8,22,25,21)(9,26,16,28)(11,27,29,24)(13,20)(14,18)(19,31)(30,32);(2,8)(3,12)(7,21)(9,24)(10,25)(11,28)(13,30)(14,15)(16,27)(17,31)(22,23)(26,29)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=gh,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=dg,e^b=efgh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #133005, #133006, #133007, … |
| maximal normal subgroup of | #133005, #133006, #133007, … |
| maximal quotient of | #124531, #124552, #124610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4799);