Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13400 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2034 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3060 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 53 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)(4,18)(5,10)(6,24)(7,11)(8,15)(9,22)(13,20)(14,29)(16,30)(17,26)(19,21)(23,32)(25,31)(27,28);(1,2)(3,13)(4,19)(5,24)(6,10)(7,11)(8,16)(9,23)(12,28)(14,27)(15,26)(17,25)(18,21)(20,29)(22,32)(30,31);(1,3,6,16,7,14,5,17)(2,8,10,27,11,25,24,13)(4,20,22,31,23,28,21,15)(9,26,18,29,19,30,32,12)Polycyclic presentation:
2^8|c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^b=eh,f^b=fh,d^c=dh,e^c=egh,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13400, #499058, #499269, … |
| maximal normal subgroup of | C2×#13400, #499058, #499269, … |
| maximal quotient of | #144716, #144717, #144752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6388);