Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | Aut(G) = C22×#7622880 (order 2048) · Inn(G) = #535 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2482 |
| prime radical | #585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 360 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:
(1,2,6,12,7,13,5,11)(3,8,15,24,16,10,22,29)(4,17,21,9,19,28,20,27)(14,25,18,26,30,32,31,23);(1,3)(2,8)(4,18,19,31)(5,22)(6,15)(7,16)(9,25,27,32)(10,13)(11,29)(12,24)(14,21,30,20)(17,23,28,26);(1,4)(2,9)(3,14)(5,21)(6,20)(7,19)(8,23)(10,26)(11,28)(12,17)(13,27)(15,31)(16,30)(18,22)(24,32)(25,29);(1,3)(2,10)(4,14)(5,22)(6,15)(7,16)(8,13)(9,26)(11,24)(12,29)(17,25)(18,21)(19,30)(20,31)(23,27)(28,32)Polycyclic presentation:
2^8|a^2=g,b^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^b=fh,e^c=eh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #508110, #508111, #508112, … |
| maximal normal subgroup of | #508110, #508111, #508112, … |
| maximal quotient of | #314610, #314611, #314612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20587);