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 | C23 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
223 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 223 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:
(2,7)(4,5)(8,15)(9,10)(11,16)(12,14)(17,18,22,19)(20,27,28,32)(21,26,31,24)(23,30,25,29);(1,2)(3,11)(4,10)(5,7)(6,9)(8,12)(13,15)(14,16)(17,19,22,18)(20,25,28,23)(21,24,31,26)(27,30,32,29);(1,3,6,13)(2,8,9,16)(4,12,5,14)(7,11,10,15)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,27)(29,31)Polycyclic presentation:
2^8|a^2=h,b^2=h,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×#10139, #236726, #236727, … |
| maximal normal subgroup of | C2×#10139, #236726, #236727, … |
| maximal quotient of | #102662, #102669, #102678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3127);