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 | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 474 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)(17,18,21,26)(19,27,29,23)(20,25,22,24)(28,32,30,31);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,29)(18,23,26,27)(20,28,22,30)(24,32,25,31);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(18,24)(20,22)(23,31)(25,26)(27,32)(28,30);(4,12)(7,14)(10,16)(13,15)(17,20,21,22)(18,25,26,24)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^8|a^2=e,b^2=e,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 | #246383, #246384, #246389, … |
| maximal normal subgroup of | #246383, #246384, #246389, … |
| maximal quotient of | #327777, #327785, #327794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22642);