Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C23 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
412 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11)(3,12,15,16)(4,14,8,13)(5,10,6,9)(17,18,23,20)(19,27,30,32)(21,26,22,25)(24,31,28,29);(1,3)(2,8)(4,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,19,23,30)(18,24,20,28)(21,29,22,31)(25,27,26,32);(1,4,7,8)(2,3,11,15)(5,16,6,12)(9,14,10,13)(17,20,23,18)(19,28,30,24)(21,25,22,26)(27,31,32,29);(1,5,7,6)(2,9,11,10)(3,13,15,14)(4,12,8,16)(17,21,23,22)(18,25,20,26)(19,29,30,31)(24,27,28,32)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=g,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,d^c=dg,e^c=eh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #153503, #249726, #249736, … |
| maximal normal subgroup of | #153503, #249726, #249736, … |
| maximal quotient of | #168133, #168138, #168165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12072);