Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#121 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
374 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 79 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 — 3 generators on 32 points:
(1,2)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=f,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=dg,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #132111, #132112, #132113, … |
| maximal normal subgroup of | #132111, #132112, #132113, … |
| maximal quotient of | #96991, #96992, #96993, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1313);