Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 73 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,22)(23,28)(24,29)(30,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=f,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,c^b=cf,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #129694, #129695, #129697, … |
| maximal normal subgroup of | #129694, #129695, #129697, … |
| maximal quotient of | #124268, #124306, #124375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4627);