Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | Aut(G) order 16384 |
| 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 | |
348 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 109 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)(3,4)(6,11)(7,16)(8,17)(9,18)(12,23)(13,19)(14,29)(15,27)(21,22)(24,34)(25,26)(28,33)(30,36)(32,35);(5,6,10,14)(7,12,16,15)(8,19,21,35)(9,23,25,27)(11,26,29,18)(13,31,32,20)(17,28,22,36)(24,33,34,30);(2,4)(5,7,10,16)(6,12,14,15)(8,17,21,22)(9,18,25,26)(11,27,29,23)(13,30,32,33)(19,28,35,36)(20,34,31,24);(5,8,10,21)(6,13,14,32)(7,17,16,22)(9,24,25,34)(11,28,29,36)(12,30,15,33)(18,20,26,31)(19,27,35,23)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dgh,e^a=eg,g^a=gh,d^b=de,e^b=eg,g^b=gh,e^d=eg,g^d=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #151513, #151517, #151567, … |
| maximal normal subgroup of | #151513, #151517, #151567, … |
| maximal quotient of | #491719, #491783, #491847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26841);