Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | Aut(G) = #72234841 (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | #1889 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,6,11,4,9,13,22)(3,12,16,28,14,7,21,24)(5,10,19,27,17,8,20,26)(15,25,29,32,18,23,30,31);(1,3,13,21,4,14,6,16)(2,7,22,28,9,12,11,24)(5,18,19,30,17,15,20,29)(8,25,26,32,10,23,27,31);(2,8)(3,14)(5,17)(7,23)(9,10)(11,27)(12,25)(16,21)(19,20)(22,26)(24,32)(28,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=egh,g^a=gh,c^b=ch,e^b=egh,d^c=dg,e^c=eh,f^c=fg,e^d=eh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152184, #152192, #152193, … |
| maximal normal subgroup of | #152184, #152192, #152193, … |
| maximal quotient of | #136823, #136824, #136849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6040);