Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| 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 | C22×C4 | C24 |
| prime radical | C2×#538 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
542 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 542 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,12,15,21)(4,9,5,18)(7,10,16,24)(8,19,23,13)(14,26,27,32)(17,25,20,29)(22,30,31,28);(1,3)(2,8)(4,13)(5,19)(6,15)(7,14)(9,21)(10,22)(11,23)(12,18)(16,27)(17,28)(20,30)(24,31)(25,32)(26,29);(1,4,16,20)(2,9,24,29)(3,13,27,30)(5,7,17,6)(8,21,31,26)(10,25,11,18)(12,22,32,23)(14,28,15,19);(2,10)(8,22)(9,25)(11,24)(12,26)(18,29)(21,32)(23,31)Polycyclic presentation:
2^8|a^2=g,c^2=gh,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 | #245078, #245079, #245082, … |
| maximal normal subgroup of | #245078, #245079, #245082, … |
| maximal quotient of | #168521, #168523, #168531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11285);