Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #1898 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1898 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,7,12,8,13,6,11)(3,5,10,23,16,24,25,14)(4,17,20,9,21,27,19,28)(15,30,26,18,31,32,29,22);(1,3,14,2)(4,15,30,9)(5,12,6,25)(7,10,24,11)(8,16,23,13)(17,20,26,22)(18,27,19,29)(21,31,32,28);(1,4,8,21)(2,9,13,28)(3,15,16,31)(5,22,24,18)(6,20,7,19)(10,29,25,26)(11,27,12,17)(14,30,23,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13103, #494757, #494759, … |
| maximal normal subgroup of | C2×#13103, #494757, #494759, … |
| maximal quotient of | #137687, #137689, #137731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6091);