Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
658 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 658 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,16)(7,17,21,27)(8,22,24,30)(9,15,10,14)(12,20,28,19)(13,25,29,23)(18,32,34,26)(31,35,36,33);(5,7,10,20,11,21,9,19)(6,12,15,27,16,28,14,17)(8,18,25,35,24,34,23,33)(13,26,30,36,29,32,22,31);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,22)(15,30)(16,29)(17,31)(19,33)(20,35)(21,34)(27,36)(28,32);(2,4)Polycyclic presentation:
2^8|a^2=h,b^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137981, #137982, #138252, … |
| maximal normal subgroup of | #137981, #137982, #138252, … |
| maximal quotient of | #347644, #347645, #347646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25931);