Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
341 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(18,22,24,27)(23,31,32,30);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,18)(19,23)(20,27)(21,24)(22,29)(25,30)(26,32)(28,31);(2,7)(4,13)(6,9)(10,12)(17,19,21,26)(18,23,24,32)(20,28,29,25)(22,30,27,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #94452, C2×#8474, #187037, … |
| maximal normal subgroup of | #94452, C2×#8474, #187037, … |
| maximal quotient of | #97334, #97338, #97344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1462);