Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2416 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
415 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 415 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:
(1,2,5,10,6,11,19,9)(3,12,15,7,17,13,16,23)(4,18,20,8,21,24,32,25)(14,28,29,26,31,22,30,27);(1,3,5,15,6,17,19,16)(2,7,9,12,11,23,10,13)(4,14,20,29,21,31,32,30)(8,22,18,26,25,28,24,27);(1,2)(3,13)(4,8)(5,9)(6,11)(7,15)(10,19)(12,17)(14,27)(16,23)(18,20)(21,25)(22,29)(24,32)(26,31)(28,30);(1,4,19,32,6,21,5,20)(2,8,10,24,11,25,9,18)(3,14,16,30,17,31,15,29)(7,22,13,27,23,28,12,26)Polycyclic presentation:
2^8|a^2=f,b^2=e,d^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501252, #501253, #501270, … |
| maximal normal subgroup of | #501252, #501253, #501270, … |
| maximal quotient of | #277852, #277856, #277858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17797);