Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7493 |
| automorphism | Aut(G) = #72286298 (order 2048) · Inn(G) = #1390 · Out(G) = C24 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1390 |
| commutator | C2×D4 | C42 |
| Frattini, prime radical, cosocle | C2×#127 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 33 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)(3,10,12,16)(4,9,14,7)(5,8)(6,11,15,13);(1,3)(2,6,8,16)(4,13)(5,11)(7,15,9,10)(12,14)Polycyclic presentation:
2^8|a^2=d,b^2=e,f^2=h,b^a=bc,c^a=cf,e^a=eh,f^a=fg,g^a=gh,c^b=ch,d^b=dfh,f^b=fh,g^b=gh,d^c=dgh,f^c=fh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7493, #151814, #151827, … |
| maximal normal subgroup of | C2×#7493, #151814, #151827, … |
| maximal quotient of | #94090, #94134, #94147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 481);