Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| 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 | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
532 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10)(3,11,14,24)(4,15,18,9)(6,8,19,23)(7,16,21,12)(13,25,27,32)(17,28,30,22)(20,29,31,26);(1,3,5,14)(2,7,10,21)(4,16,18,12)(6,13,19,27)(8,20,23,31)(9,24,15,11)(17,29,30,26)(22,32,28,25);(4,17)(9,22)(11,25)(12,26)(15,28)(16,29)(18,30)(24,32);(2,8)(7,20)(9,22)(10,23)(11,25)(15,28)(21,31)(24,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #245074, #245075, #245077, … |
| maximal normal subgroup of | #245074, #245075, #245077, … |
| maximal quotient of | #168500, #168510, #168562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11275);