Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10386 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
207 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,12)(4,15)(8,24,16,20)(10,23,26,27)(13,30)(17,18)(22,25,29,32)(28,31);(2,8,9,16)(3,11)(7,20,12,24)(10,25,26,32)(13,28)(14,21)(22,27,29,23)(30,31);(1,2,6,10)(3,12,11,27)(4,16,18,32)(5,9,19,26)(7,21,23,14)(8,15,25,17)(13,22,28,20)(24,31,29,30)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=h,e^2=g,f^2=g,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,e^b=eh,d^c=dg,e^c=eg,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | #151460, #151461, C2×#10387, … |
| maximal normal subgroup of | #151460, #151461, C2×#10387, … |
| maximal quotient of | #104731, #104732, #104739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3375);