Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10128 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
236 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 51 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 — 3 generators on 32 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,23)(19,27)(20,21)(25,26)(28,31)(29,30);(4,13)(7,12)(9,16)(11,14)(17,18)(19,28)(20,25)(21,23)(22,26)(24,27)(29,32)(30,31);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(17,19,22,30)(18,24,26,32)(20,27,21,29)(23,28,25,31)Polycyclic presentation:
2^8|a^2=gh,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,e^b=egh,f^b=fg,d^c=dh,e^c=eh,f^c=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10141, #236716, #236728, … |
| maximal normal subgroup of | C2×#10141, #236716, #236728, … |
| maximal quotient of | #102655, #102674, #102676, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3129);