Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| 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 | #3480 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
469 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18,23,28)(19,29,31,25)(20,32,30,24)(21,27,22,26);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,24)(20,21)(22,30)(23,31)(25,26)(27,29)(28,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,30)(18,25,28,29)(19,21,31,22)(24,26,32,27);(17,21,23,22)(18,26,28,27)(19,20,31,30)(24,25,32,29)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,e^b=eg,f^b=fh,e^c=eh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #270633, #270634, #270657, … |
| maximal normal subgroup of | #270633, #270634, #270657, … |
| maximal quotient of | #166017, #166024, #166029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11779);