Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(2,5)(6,8)(17,19,20,26,21,25,27,31)(18,22,24,29,23,28,30,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,25)(22,28)(26,31)(29,32);(9,11,12,15)(10,14,13,16)(18,23)(22,28)(24,30)(29,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=e,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #135502, #135505, C2×#21433, … |
| maximal normal subgroup of | #135502, #135505, C2×#21433, … |
| maximal quotient of | #353842, #353989, #354129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14421);