Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
424 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,11)(7,13)(8,12)(10,15)(14,16)(17,18,23,20)(19,27,30,32)(21,26,22,25)(24,31,28,29);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19,23,30)(18,24,20,28)(21,29,22,31)(25,27,26,32);(1,3,5,9)(2,7,8,14)(4,10,11,15)(6,13,12,16)(17,20,23,18)(19,28,30,24)(21,25,22,26)(27,31,32,29);(3,9)(7,14)(10,15)(13,16)(17,21,23,22)(18,25,20,26)(19,29,30,31)(24,27,28,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #250427, #250429, #250432, … |
| maximal normal subgroup of | #250427, #250429, #250432, … |
| maximal quotient of | #169654, #169672, #169717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11430);