Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| 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 | |
529 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 529 conjugacy classes of subgroups — too many to draw. Only its 117 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)(3,12,15,25)(4,16,19,9)(6,11,20,8)(7,17,23,13)(14,27,30,28)(18,24,32,26)(21,29,22,31);(1,3,5,15)(2,7,10,23)(4,17,19,13)(6,14,20,30)(8,21,11,22)(9,25,16,12)(18,31,32,29)(24,27,26,28);(4,18)(9,26)(12,27)(13,29)(16,24)(17,31)(19,32)(25,28);(2,8)(7,21)(9,24)(10,11)(12,28)(16,26)(22,23)(25,27)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #245158, #245159, #245161, … |
| maximal normal subgroup of | #245158, #245159, #245161, … |
| maximal quotient of | #168504, #168527, #168565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11307);