Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | Aut(G) = C22×#501097 (order 2048) · Inn(G) = #353 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #353 |
| commutator | C42 | C42 |
| Frattini, cosocle | C2×#124 | C22 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 145 conjugacy classes of subgroups — too many to draw. Only its 35 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:
(5,6,7,13)(8,14,18,11)(9,12,17,19)(10,15,16,20);(1,2,3,4)(5,7,8,17,10,16,9,18)(6,11,14,13,15,19,12,20)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=g,e^2=h,f^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fg,g^a=gh,c^b=ch,d^b=dfgh,d^c=dgh,e^d=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7508, #151866, #151867, … |
| maximal normal subgroup of | C2×#7508, #151866, #151867, … |
| maximal quotient of | #94107, #94158, #94159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 496);