Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | Aut(G) = #72256639 (order 2048) · Inn(G) = #410 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #410 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | #1782 |
| prime radical | #3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C42 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 69 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 — 3 generators on 32 points:
(1,2)(3,12)(4,17,19,9)(5,10)(6,11)(7,14,23,13)(8,18,25,32)(15,24)(16,28)(20,26)(21,31)(22,30,27,29);(1,3)(2,7)(4,14)(5,15)(6,16)(8,24)(9,12)(10,22)(11,23)(13,19)(17,28)(18,30)(20,31)(21,25)(26,27)(29,32);(1,4)(2,8,11,25)(3,13)(5,18)(6,19)(7,21,23,24)(9,10,17,26)(12,27,28,22)(14,16)(15,29)(20,32)(30,31)Polycyclic presentation:
2^8|a^2=f,c^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch,f^b=fg,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152080, #152081, #152089, … |
| maximal normal subgroup of | #152080, #152081, #152089, … |
| maximal quotient of | #129288, #129294, #129370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5708);