Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
517 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 179 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 — 4 generators on 32 points:
(1,2,4,10,6,9,16,23)(3,8,13,22,11,21,24,32)(5,7,15,19,12,18,26,30)(14,17,28,29,25,20,27,31);(2,7)(3,11)(4,15)(8,20)(9,18)(13,27)(14,25)(16,26)(17,21)(22,32)(24,28)(29,31);(1,3,12,25)(2,8,18,20)(4,13,26,27)(5,14,6,11)(7,17,9,21)(10,22,30,31)(15,28,16,24)(19,29,23,32);(2,9)(7,18)(8,21)(10,23)(17,20)(19,30)(22,32)(29,31)Polycyclic presentation:
2^8|a^2=f,c^2=gh,f^2=h,b^a=be,d^a=dh,e^a=eg,c^b=ch,f^b=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #107170, #107184, #108278, … |
| maximal normal subgroup of | #107170, #107184, #108278, … |
| maximal quotient of | #162580, #162585, #162590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10386);