Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
495 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 183 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 — 4 generators on 32 points:
(2,7,9,22)(3,12)(4,16)(8,24,25,30)(10,23,11,17)(13,15)(18,20)(21,27,32,26);(1,2,5,10)(3,8,12,26)(4,17,16,22)(6,11,19,9)(7,20,23,18)(13,25,15,27)(14,30,28,32)(21,29,24,31);(1,3)(2,8)(4,14)(5,12)(6,15)(7,21)(9,25)(10,26)(11,27)(13,19)(16,28)(17,30)(18,31)(20,29)(22,32)(23,24);(3,13)(8,25)(12,15)(14,29)(21,32)(24,30)(26,27)(28,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgh49 conjugacy classes — view the full character table.
| maximal subgroup of | #253509, #253515, #253521, … |
| maximal normal subgroup of | #253509, #253515, #253521, … |
| maximal quotient of | #160636, #160637, #160898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10609);