Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | Aut(G) = C22×#6346637 (order 2048) · Inn(G) = #327 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #327 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C22×M16 | C22 |
| prime radical | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(6,10)(7,14)(8,16)(11,15)(12,19)(13,18)(17,20);(5,6)(7,15,17,19)(8,12)(9,13)(10,14,18,20)(11,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7254, #127504, #127585, … |
| maximal normal subgroup of | C2×#7254, #127504, #127585, … |
| maximal quotient of | #93031, #93033, #93039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 242);