Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | Aut(G) = #72313686 (order 4096) · Inn(G) = #447 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#447 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
566 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9)(5,17)(6,21)(7,11)(8,14)(10,20)(13,29)(15,28)(16,27)(18,23)(19,31)(22,26)(24,30)(25,32);(1,3)(2,8)(4,13)(5,18)(6,22)(7,15)(9,23)(10,26)(11,24)(12,17)(14,20)(16,30)(19,32)(21,25)(27,29)(28,31);(5,19)(10,27)(12,28)(14,30)(16,20)(17,31)(18,32)(26,29);(1,4,7,6)(2,9,11,21)(3,13,15,22)(5,16,19,20)(8,23,24,25)(10,17,27,31)(12,29,28,26)(14,18,30,32)Polycyclic presentation:
2^8|d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,e^b=eg,f^b=fh,g^b=gh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #137391, #137394, #137399, … |
| maximal normal subgroup of | #137391, #137394, #137399, … |
| maximal quotient of | #484193, #484194, #484195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26518);