Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| 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 | |
537 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 537 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(1,2,7,11)(3,12,14,27)(4,9,17,26)(5,18,20,29)(6,10,16,25)(8,21,24,13)(15,22,28,32)(19,23,30,31);(1,3)(2,8)(4,13)(5,19)(6,15)(7,14)(9,22)(10,23)(11,24)(12,18)(16,28)(17,21)(20,30)(25,31)(26,32)(27,29);(1,4,16,20)(2,9,25,29)(3,13,28,30)(5,7,17,6)(8,22,31,27)(10,18,11,26)(12,24,32,23)(14,21,15,19);(2,10)(8,23)(9,18)(11,25)(12,22)(24,31)(26,29)(27,32)Polycyclic presentation:
2^8|a^2=h,c^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #246353, #246354, #246357, … |
| maximal normal subgroup of | #246353, #246354, #246357, … |
| maximal quotient of | #165888, #165891, #165899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11137);