Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,21,25,28,31)(19,27,29,24,20,23,30,32);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,28,30,21,20,22,29)(18,23,31,24,25,27,26,32);(10,14)(13,16)(17,20,21,19)(18,24,25,32)(22,29,28,30)(23,26,27,31);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=egh,b^2=ef,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #311166, #311174, #311182, … |
| maximal normal subgroup of | #311166, #311174, #311182, … |
| maximal quotient of | #360456, #363087, #366420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20812);