Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18524 |
| 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 | C42 | C24 |
| prime radical | #3482 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
461 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 461 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,6,10)(3,11,13,16)(4,9,5,8)(7,14,15,12)(17,18,23,28)(19,25,30,31)(20,24,29,32)(21,27,22,26);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19,23,30)(18,24,28,32)(20,21,29,22)(25,26,31,27);(17,20)(18,25)(19,21)(22,30)(23,29)(24,26)(27,32)(28,31);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(17,21,23,22)(18,26,28,27)(19,29,30,20)(24,31,32,25)Polycyclic presentation:
2^8|a^2=g,b^2=h,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,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #272435, #272436, #272437, … |
| maximal normal subgroup of | #272435, #272436, #272437, … |
| maximal quotient of | #165887, #165931, #165954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11526);