Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| 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 | #2635 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
498 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 498 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,15)(8,21)(9,19)(10,16)(12,27)(13,29)(14,26)(17,31)(20,22)(23,32)(24,28)(25,30);(1,2,5,10)(3,12,14,8)(4,16,15,7)(6,9,18,23)(11,24,26,22)(13,20,30,28)(17,32,31,19)(21,29,27,25);(4,17)(7,19)(13,25)(15,31)(16,32)(20,21)(27,28)(29,30);(1,3,5,14)(2,8,10,12)(4,13,15,30)(6,11,18,26)(7,20,16,28)(9,22,23,24)(17,25,31,29)(19,21,32,27)Polycyclic presentation:
2^8|b^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #245616, #245623, #245628, … |
| maximal normal subgroup of | #245616, #245623, #245628, … |
| maximal quotient of | #168915, #168927, #168971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11340);