Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) = #72282918 (order 2048) · Inn(G) = #442 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #442 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1762 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
206 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,8,13,7,23)(3,14,17,29,18,30,21,25)(4,10,6,22,20,28,19,11)(9,15,24,16,27,31,26,32);(1,3,6,15,5,21,20,32,8,18,19,31,7,17,4,16)(2,9,10,25,23,24,11,14,13,27,28,29,12,26,22,30);(1,4,5,6,8,20,7,19)(2,10,12,22,13,28,23,11)(3,15,17,16,18,31,21,32)(9,14,24,29,27,30,26,25)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=gh,g^2=h,b^a=bd,d^a=dg,f^a=fg,g^a=gh,c^b=cg,e^b=eh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12651, #493088, #493247, … |
| maximal normal subgroup of | C2×#12651, #493088, #493247, … |
| maximal quotient of | #128557, #128574, #128608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5639);