Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | Aut(G) = #72285761 (order 4096) · Inn(G) = #353 · Out(G) = D42 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #353 |
| commutator | C42 | C42 |
| Frattini, prime radical, cosocle | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
174 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 174 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) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6)(7,14,15,10)(8,12,17,11)(9,13)(16,18,20,19);(1,2,3,4)(5,7,8,15,9,16,17,20)(6,10,11,14,13,19,12,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=gh,e^2=h,f^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fg,g^a=gh,d^b=dfg,d^c=dgh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94642, #94643, #94654, … |
| maximal normal subgroup of | #94642, #94643, #94654, … |
| maximal quotient of | #94101, #94153, #94177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 486);