Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2631 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
502 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 502 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)(4,16)(8,21)(9,19)(10,17)(12,27)(14,30)(15,26)(18,29)(20,23)(22,32)(24,28)(25,31);(1,2,5,10)(3,12,15,8)(4,17,16,7)(6,9,13,22)(11,24,26,23)(14,20,31,28)(18,32,29,19)(21,30,27,25);(4,18)(7,19)(14,25)(16,29)(17,32)(20,21)(27,28)(30,31);(1,3,13,26)(2,8,22,24)(4,14,29,30)(5,15,6,11)(7,20,32,27)(9,23,10,12)(16,31,18,25)(17,28,19,21)Polycyclic presentation:
2^8|b^2=g,d^2=gh,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 | #245904, #245911, #245931, … |
| maximal normal subgroup of | #245904, #245911, #245931, … |
| maximal quotient of | #168938, #168994, #169050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11347);