Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 629 conjugacy classes of subgroups — too many to draw. Only its 127 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 — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,13,22,28)(7,15,17,29,18,11,21,24)(8,12,20,27,19,26,32,36)(16,25,31,35,30,23,33,34);(5,7,10,18)(6,11,13,15)(8,16,19,30)(9,21,22,17)(12,25,26,23)(14,29,28,24)(20,33,32,31)(27,34,36,35);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,26)(14,27)(15,25)(17,31)(18,30)(21,33)(22,32)(24,34)(28,36)(29,35)Polycyclic presentation:
2^8|a^2=g,b^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,g^b=gh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #133415, #133417, #133419, … |
| maximal normal subgroup of | #133415, #133417, #133419, … |
| maximal quotient of | #111469, #111470, #111546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3962);