Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| 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 | |
337 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 337 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)(7,15)(8,20)(9,23)(12,16)(13,32)(14,28)(18,19)(21,25)(22,34)(24,26)(27,33)(30,36)(31,35);(5,6,10,14)(7,16,19,35)(8,21,24,32)(9,25,23,13)(11,26,28,20)(12,29,31,17)(15,27,18,36)(22,33,34,30);(2,4);(5,7,10,19)(6,12,14,31)(8,22,24,34)(9,15,23,18)(11,27,28,36)(13,33,25,30)(16,32,35,21)(17,26,29,20)Polycyclic presentation:
2^8|b^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 | #513167, #513170, #513175, … |
| maximal normal subgroup of | #513167, #513170, #513175, … |
| maximal quotient of | #491715, #491726, #491779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26839);