Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) = C2×#72267396 (order 2048) · Inn(G) = C2×C42 · Out(G) = C26 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 97 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 — 3 generators on 20 points:
(1,2,3,4)(5,6,9,14,8,13,17,20,10,15,18,7,12,11,19,16);(2,4)(6,11,15,13)(7,16,14,20)(9,18)(17,19);(5,7,17,6,12,20,9,11,10,14,19,15,8,16,18,13)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=h,e^2=g,g^2=h,b^a=bd,f^a=fh,c^b=cg,e^b=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128702, #128703, #128715, … |
| maximal normal subgroup of | #128702, #128703, #128715, … |
| maximal quotient of | #129049, #129050, #129061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5654);