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 | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
228 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 228 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,3)(4,11,12,16)(5,10,15,8)(7,14,9,13)(17,18)(19,26)(20,23)(21,25)(22,28)(24,30)(27,32)(29,31);(1,3,6,2)(4,9,12,7)(5,8,15,10)(11,14,16,13)(17,19,21,29)(18,22,25,32)(20,27,30,28)(23,26,24,31);(1,4)(2,7)(3,9)(5,14)(6,12)(8,16)(10,11)(13,15)(18,23)(19,27)(20,30)(24,25)(26,31)(28,29)Polycyclic presentation:
2^8|a^2=gh,b^2=h,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×#10142, #236726, #236728, … |
| maximal normal subgroup of | C2×#10142, #236726, #236728, … |
| maximal quotient of | #102661, #102668, #102681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3130);