Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| 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 | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
459 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 459 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(18,22,24,27)(23,29,30,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,18)(19,23)(20,27)(21,24)(22,28)(25,30)(26,32)(29,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,25)(23,30)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=g,e^2=h,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dgh61 conjugacy classes — view the full character table.
| maximal subgroup of | #221147, #221149, #221154, … |
| maximal normal subgroup of | #221147, #221149, #221154, … |
| maximal quotient of | #160516, #160517, #160522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10604);