Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
344 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 107 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:
(2,6)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(2,9)(3,10)(6,13)(11,14)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^8|b^2=eg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dgh,d^c=dg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #285371, #285373, #285388, … |
| maximal normal subgroup of | #285371, #285373, #285388, … |
| maximal quotient of | #189163, #197902, #197984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12542);