Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) order 16384 |
| 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 | |
487 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 487 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)(3,9,11,6)(4,8)(5,7)(10,15,16,13)(12,14)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(1,3,5,11)(2,6,7,9)(4,10,12,16)(8,13,14,15)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(2,7)(4,12)(6,9)(10,16)(18,23)(22,31)(24,28)(25,32);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=gh,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 | #221191, #221192, #221195, … |
| maximal normal subgroup of | #221191, #221192, #221195, … |
| maximal quotient of | #160638, #160639, #160900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10615);