Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(6,11)(8,18)(9,20)(12,24)(13,28)(14,25)(16,31)(17,30)(19,22)(21,23)(26,35)(27,36)(29,34)(32,33);(5,6,10,14)(7,12,15,26)(8,19,21,28)(9,22,20,13)(11,23,25,18)(16,32,29,36)(17,33,30,27)(24,34,35,31);(1,3,4,2)(5,7)(6,12)(8,16)(9,17)(10,15)(11,24)(13,27)(14,26)(18,31)(19,32)(20,30)(21,29)(22,33)(23,34)(25,35)(28,36);(7,15)(12,26)(16,29)(17,30)(24,35)(27,33)(31,34)(32,36)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,e^2=gh,g^2=h,b^a=be,e^a=eg,g^a=gh,e^b=eg,g^b=gh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #498165, #498167, #498169, … |
| maximal normal subgroup of | #498165, #498167, #498169, … |
| maximal quotient of | #483503, #483789, #483791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26443);