Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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:
(2,8)(3,13)(4,17)(5,6)(9,10)(11,27)(12,26)(14,31)(15,29)(16,22)(18,23)(20,21)(24,25)(28,30);(1,2,7,12)(3,14,16,9)(4,18,21,27)(5,11,6,23)(8,20,26,17)(10,29,31,15)(13,28,22,25)(19,30,32,24);(1,3)(2,9)(4,15)(5,22)(6,13)(7,16)(8,24)(10,27)(11,28)(12,14)(17,19)(18,31)(20,32)(21,29)(23,25)(26,30);(1,3)(2,10)(4,19)(5,13)(6,22)(7,16)(8,25)(9,27)(11,30)(12,31)(14,18)(15,17)(20,29)(21,32)(23,24)(26,28)Polycyclic presentation:
2^8|b^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gh,c^b=ch,d^b=de,e^b=eg,g^b=gh,f^c=fh,e^d=eg,g^d=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151524, #151526, #151580, … |
| maximal normal subgroup of | #151524, #151526, #151580, … |
| maximal quotient of | #491729, #491734, #491744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26846);