Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
406 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(2,8)(4,17)(5,6)(9,22)(10,25)(11,12)(13,24)(14,28)(15,20)(18,19)(27,31)(29,30);(1,2,6,12,7,11,5,8)(3,13,15,24,16,22,20,9)(4,10,18,26,19,25,17,21)(14,27,28,32,29,31,30,23);(1,3)(2,9)(4,14)(5,15)(6,20)(7,16)(8,13)(10,23)(11,24)(12,22)(17,28)(18,30)(19,29)(21,27)(25,32)(26,31);(1,4,6,18,7,19,5,17)(2,10,12,26,11,25,8,21)(3,14,20,30,16,29,15,28)(9,23,22,31,24,32,13,27)Polycyclic presentation:
2^8|b^2=g,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #500749, #500751, #500753, … |
| maximal normal subgroup of | #500749, #500751, #500753, … |
| maximal quotient of | #316992, #316995, #316998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21171);