Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9431 |
| 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 | |
226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(2,6)(3,10)(4,7)(9,14)(11,12)(13,15)(17,18,23,26,21,20,22,25)(19,27,29,31,30,24,28,32);(1,2,7,14,5,9,4,6)(3,8,10,15,12,16,11,13)(17,19,22,29,21,30,23,28)(18,24,26,31,20,27,25,32);(1,3)(2,8)(4,11)(5,12)(6,13)(7,10)(9,16)(14,15)(17,20,21,18)(19,27,30,24)(22,25,23,26)(28,31,29,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94772, C2×#9438, #200965, … |
| maximal normal subgroup of | #94772, C2×#9438, #200965, … |
| maximal quotient of | #100116, #100122, #100871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2426);