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 | |
214 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 214 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)(3,13,16,29)(4,10)(5,21)(6,23)(7,12)(8,15,25,30)(9,20)(11,22)(14,26,18,32)(17,24,31,27)(19,28);(1,3,5,17,22,14,19,30,7,16,20,31,6,18,4,15)(2,8,10,26,23,24,9,13,12,25,28,32,11,27,21,29);(3,14,16,18)(4,19)(5,20)(8,24,25,27)(9,21)(10,28)(13,26,29,32)(15,31,30,17)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=g,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×#12649, #493082, #493247, … |
| maximal normal subgroup of | C2×#12649, #493082, #493247, … |
| maximal quotient of | #128555, #128556, #128572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5637);