Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | Aut(G) order 8192 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
448 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 448 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,5)(9,10)(12,13)(14,15)(18,23)(20,21)(24,31)(25,28)(26,29)(30,32);(1,2)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(17,18,22,25)(19,24,27,32)(20,23,21,28)(26,30,29,31);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,30)(25,32)(28,31);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,15,16,14)(17,20,22,21)(18,23,25,28)(19,26,27,29)(24,30,32,31)Polycyclic presentation:
2^8|b^2=h,d^2=g,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,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153493, #347444, #347445, … |
| maximal normal subgroup of | #153493, #347444, #347445, … |
| maximal quotient of | #165886, #165931, #165953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11992);