Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = C4×#60399 (order 1024) · Inn(G) = #99 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #99 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C2×#131 | C22 |
| socle | C2 | #1282 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
124 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,13,26,36,8,12,6,27,25,37,28,53)(3,14,16,38,17,41,45,61,18,40,23,39,46,60,50,62)(4,19,20,34,21,47,48,59,22,10,24,33,49,35,51,58)(9,29,30,43,32,55,56,44,31,42,52,54,57,63,64,15);(1,3,6,16,7,17,28,45,8,18,5,23,25,46,26,50)(2,9,11,30,13,32,36,56,12,31,27,52,37,57,53,64)(4,15,20,42,21,43,48,63,22,44,24,29,49,54,51,55)(10,14,34,39,35,41,59,62,19,40,33,38,47,60,58,61)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=g,e^2=g,g^2=h,b^a=bc,c^a=cf,e^a=eh,f^a=fh,d^b=dfh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7208, #127084, #127133, … |
| maximal normal subgroup of | C2×#7208, #127084, #127133, … |
| maximal quotient of | #92997, #92998, #92999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 196);