Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #1790 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3540 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
379 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 379 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,6,11,7,12,4,9)(3,13,17,28,16,29,14,30)(5,19,18,26,20,10,21,25)(8,22,23,15,24,31,27,32);(1,3)(2,8)(4,14)(5,15)(6,17)(7,16)(9,23)(10,13)(11,27)(12,24)(18,31)(19,29)(20,32)(21,22)(25,30)(26,28);(1,4,7,6)(2,9,12,11)(3,14,16,17)(5,18,20,21)(8,23,24,27)(10,26,19,25)(13,28,29,30)(15,31,32,22)Polycyclic presentation:
2^8|a^2=f,c^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^b=egh,f^b=fg,d^c=dg,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12756, #493569, #493570, … |
| maximal normal subgroup of | C2×#12756, #493569, #493570, … |
| maximal quotient of | #129600, #129603, #129633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5744);