Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,6,10,5,8)(3,11,13,7,12,16,14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,3,5,14,6,12,4,13)(2,7,9,16,10,15,8,11)(17,19)(18,22,24,25)(20,26)(21,27)(23,29,30,31)(28,32);(3,12)(7,15)(11,16)(13,14)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #107312, #107313, #107519, … |
| maximal normal subgroup of | #107312, #107313, #107519, … |
| maximal quotient of | #95321, #95323, #95327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1044);