Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11753 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | #392 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
223 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 223 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)(3,13)(4,17,20,7)(5,11)(6,12)(8,14)(9,18,23,32)(10,21)(15,27,19,24)(16,25)(22,31,28,29)(26,30);(2,7)(3,14)(4,18)(8,24)(9,11)(10,23)(12,17)(13,28)(16,30)(20,32)(22,25)(26,27);(1,3,6,16)(2,8,12,26)(4,19,20,15)(5,14,21,30)(7,22,17,28)(9,27,23,24)(10,25,11,13)(18,31,32,29)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efg,d^c=dh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132093, #132095, #132096, … |
| maximal normal subgroup of | #132093, #132095, #132096, … |
| maximal quotient of | #124340, #124362, #124381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4828);