Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×H(3,𝔽4) | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
533 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 533 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 4 generators on 32 points:
(1,2,6,12)(3,7,14,22)(4,8,16,25)(5,9,18,26)(10,27,28,13)(11,19,23,17)(15,31,30,20)(21,29,24,32);(2,7)(3,13)(4,16)(8,20)(9,10)(11,24)(12,22)(14,27)(15,32)(17,19)(21,23)(25,31)(26,28)(29,30);(2,8)(3,14)(4,17)(7,20)(9,23)(10,21)(11,26)(12,25)(13,27)(15,29)(16,19)(22,31)(24,28)(30,32);(2,9)(7,10)(8,23)(11,25)(12,26)(20,21)(22,28)(24,31)Polycyclic presentation:
2^8|a^2=h,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153201, #153203, #245904, … |
| maximal normal subgroup of | #153201, #153203, #245904, … |
| maximal quotient of | #165878, #165901, #165911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11985);