Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #380 |
| prime radical | C23 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
189 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 77 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:
(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,32)(30,31);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)(18,22)(23,31)(24,29)(25,32);(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25)Polycyclic presentation:
2^8|a^2=f,b^2=dh,c^2=h,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131748, #131749, #131756, … |
| maximal normal subgroup of | #131748, #131749, #131756, … |
| maximal quotient of | #98861, #98863, #98885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1649);