Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
267 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,5,12,17,29,3,15)(4,20,23,30,21,9,18,26)(6,13,16,8,7,14,19,10)(11,31,28,22,27,32,25,24);(2,8,13,12)(3,16)(4,21)(5,19)(9,25,11,30)(10,14,15,29)(18,31)(20,28,27,26)(22,24)(23,32);(2,9)(3,5)(4,22)(8,25)(10,28)(11,13)(12,30)(14,27)(15,26)(16,19)(18,32)(20,29)(21,24)(23,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153305, C2×#9425, #200838, … |
| maximal normal subgroup of | #153305, C2×#9425, #200838, … |
| maximal quotient of | #100091, #100101, #100771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2413);