Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| 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 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
635 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 635 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:
(5,6)(7,15)(8,19,21,12)(9,22)(10,14)(11,17)(13,24)(16,30,32,29)(18,28)(20,27,36,34)(23,26)(25,31,33,35);(1,2)(3,4)(5,7,10,18)(6,11,14,26)(8,16,21,32)(9,23,24,17)(12,25,19,33)(13,28,22,15)(20,35,36,31)(27,30,34,29);(2,4)(5,8)(6,12)(7,16)(9,20)(10,21)(11,25)(13,27)(14,19)(15,29)(17,31)(18,32)(22,34)(23,35)(24,36)(26,33)(28,30)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #133383, #133385, #133387, … |
| maximal normal subgroup of | #133383, #133385, #133387, … |
| maximal quotient of | #111234, #111235, #111552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3973);