Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
475 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 475 conjugacy classes of subgroups — too many to draw. Only its 187 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 — 4 generators on 32 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,26)(20,25)(21,29)(22,24)(23,27)(28,32)(30,31);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,22,28)(18,23,24,31)(20,27,21,30)(25,32,29,26);(18,24)(23,31)(25,29)(26,32);(17,20,22,21)(18,25,24,29)(19,27,28,30)(23,32,31,26)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=h,f^2=gh,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #106019, #106026, #106055, … |
| maximal normal subgroup of | #106019, #106026, #106055, … |
| maximal quotient of | #160535, #160549, #160554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10727);