Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#540 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
409 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 71 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,26,28,23)(20,22)(21,25)(24,29)(27,31,32,30);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(18,22)(20,29)(23,30)(24,25)(26,31)(27,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,21,28)(18,23,25,26)(20,27,29,32)(22,30,24,31)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fgh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94526, #94528, #132093, … |
| maximal normal subgroup of | #94526, #94528, #132093, … |
| maximal quotient of | #97018, #97024, #97029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1506);