Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 417 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(2,4)(5,6)(7,15)(8,20)(9,12)(10,23)(11,18)(13,16)(14,21)(17,27)(19,34)(22,35)(24,32)(25,36)(26,31)(28,30)(29,33);(5,7,16,30,10,19,22,32,9,17,31,36,21,33,8,18)(6,11,20,29,14,25,26,27,12,24,35,34,23,28,13,15);(1,2,3,4)(6,12)(11,24)(13,26)(14,23)(15,27)(20,35)(25,28)(29,34);(7,17)(11,24)(15,27)(18,32)(19,33)(25,28)(29,34)(30,36)Polycyclic presentation:
2^8|b^2=eg,c^2=fh,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137543, #143211, #143216, … |
| maximal normal subgroup of | #137543, #143211, #143216, … |
| maximal quotient of | #491437, #491438, #491458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26707);