Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 366 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)(3,13,16,27)(4,10,5,18)(6,11)(7,12)(8,14)(9,17,26,20)(15,29,32,22)(19,24)(21,25)(23,30)(28,31);(1,3,7,16)(2,8,12,24)(4,14,5,19)(6,15,21,32)(9,22,26,29)(10,27,18,13)(11,28,25,23)(17,30,20,31);(1,4)(2,9)(3,14)(5,7)(6,20)(8,22)(10,25)(11,18)(12,26)(13,28)(15,31)(16,19)(17,21)(23,27)(24,29)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=eg,f^b=fgh,d^c=dgh,e^c=eh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152563, #152566, #152590, … |
| maximal normal subgroup of | #152563, #152566, #152590, … |
| maximal quotient of | #129586, #129590, #129628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5767);