Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
453 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18)(19,24)(20,29)(21,26)(22,31)(23,27)(25,30)(28,32);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,24)(20,23)(21,28)(22,30)(25,31)(26,32)(27,29);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(4,12)(7,14)(10,16)(13,15)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=de,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #246385, #246386, #246387, … |
| maximal normal subgroup of | #246385, #246386, #246387, … |
| maximal quotient of | #327778, #327786, #327793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22643);