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)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,16)(12,14)(20,29)(23,28)(25,32)(27,30);(1,3,6,11)(2,8,9,16)(4,12,5,10)(7,14,13,15)(17,19)(18,22)(20,30)(21,26)(23,32)(24,31)(25,28)(27,29);(1,4,6,5)(2,7,9,13)(3,10,11,12)(8,15,16,14)Polycyclic presentation:
2^8|a^2=h,c^2=g,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 | #249810, #249820, #249840, … |
| maximal normal subgroup of | #249810, #249820, #249840, … |
| maximal quotient of | #168126, #168152, #168217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11713);