Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| 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 | #3551 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 376 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:
(1,2,5,10,6,8,18,23)(3,11,14,27,13,28,31,32)(4,15,17,26,16,9,19,24)(7,20,22,12,21,29,25,30);(1,3,6,13)(2,7,8,21)(4,12,16,30)(5,14,18,31)(9,11,15,28)(10,25,23,22)(17,29,19,20)(24,32,26,27);(2,8)(5,18)(7,21)(14,31)(17,19)(20,29)(24,26)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=egh,f^b=fg,d^c=dg,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12770, #493600, #493601, … |
| maximal normal subgroup of | C2×#12770, #493600, #493601, … |
| maximal quotient of | #129576, #129579, #129618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5758);