Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 107 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)(5,6,10,15)(7,16,17,28)(8,21,24,30)(9,12,23,14)(11,26,27,19)(13,31,33,22)(18,34,35,29)(20,36,25,32);(5,7)(6,11)(8,18)(9,25)(10,17)(12,29)(13,32)(14,34)(15,27)(16,21)(19,22)(20,23)(24,35)(26,31)(28,30)(33,36);(2,4)(7,17)(11,27)(16,28)(18,35)(19,26)(20,25)(29,34)(32,36);(6,12)(7,18)(8,22)(9,23)(11,28)(14,15)(16,27)(17,35)(19,20)(21,30)(24,31)(25,26)(29,32)(34,36)Polycyclic presentation:
2^8|a^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gh,c^b=ch,d^b=de,e^b=eg,g^b=gh,e^d=eg,g^d=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151520, #151522, #151574, … |
| maximal normal subgroup of | #151520, #151522, #151574, … |
| maximal quotient of | #491760, #491773, #491824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26827);