Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
421 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,12)(5,10)(6,9)(7,8)(13,16)(14,15)(17,18,21,23)(19,25,27,32)(20,28,30,24)(22,29,31,26);(1,3)(2,4)(5,15)(6,12)(7,13)(8,14)(9,16)(10,11)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,4)(2,3)(5,11)(6,16)(7,14)(8,13)(9,12)(10,15)(17,18)(19,22)(20,28)(21,23)(24,30)(25,29)(26,32)(27,31);(2,8)(4,14)(9,10)(11,16)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19085, #6351501, #6351502, … |
| maximal normal subgroup of | C2×#19085, #6351501, #6351502, … |
| maximal quotient of | #168134, #168144, #168146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12073);