Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | Aut(G) order 16384 |
| 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 | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
438 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 438 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:
(2,7)(4,5)(8,14)(9,13)(10,12)(15,16)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,28,32,29);(1,2,6,9)(3,8,11,16)(4,7,5,13)(10,15,12,14)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,3)(2,8)(4,12)(5,10)(6,11)(7,14)(9,16)(13,15)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29);(1,4,6,5)(2,7,9,13)(3,10,11,12)(8,15,16,14)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)Polycyclic presentation:
2^8|a^2=h,b^2=gh,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #249808, #249838, #249858, … |
| maximal normal subgroup of | #249808, #249838, #249858, … |
| maximal quotient of | #168136, #168163, #168195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11723);