Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C22×C4 | C42 |
| Frattini, prime radical, cosocle | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,7,12,6)(4,10)(5,9)(8,13)(11,14,16,15);(2,6,8,14)(3,11)(7,9,15,10)(12,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fh,c^b=cg,d^b=dfh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | #93213, #93214, #93237, … |
| maximal normal subgroup of | #93213, #93214, #93237, … |
| maximal quotient of | #93109, #93113, #93116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 295);