Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11757 |
| automorphism | Aut(G) order 4096 |
| 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 | #392 | C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 209 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:
(1,2,4,10,5,11,17,28)(3,9,14,26,13,21,16,27)(6,8,18,24,19,25,15,30)(7,20,22,29,23,31,32,12);(2,7)(3,12)(4,15)(8,9)(10,26)(11,23)(13,29)(14,16)(17,18)(20,31)(21,25)(22,24)(27,28)(30,32);(2,8)(3,13)(7,21)(9,23)(10,24)(11,25)(12,29)(14,16)(20,31)(22,27)(26,32)(28,30)Polycyclic presentation:
2^8|a^2=e,d^2=g,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cfh,d^b=dg,e^b=efg,d^c=dh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132111, #132112, #132114, … |
| maximal normal subgroup of | #132111, #132112, #132114, … |
| maximal quotient of | #124344, #124356, #124386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4839);