Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | Aut(G) = #72317053 (order 4096) · Inn(G) = #486 · Out(G) = #392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #486 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2041 |
| prime radical | #3150 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
221 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 55 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)(3,6)(4,17)(5,24)(7,16)(8,13)(9,15)(10,20)(11,26)(12,25)(14,23)(18,27)(19,32)(21,30)(22,31)(28,29);(1,3)(2,9)(4,18)(5,12)(6,24)(7,11)(8,16)(10,22)(13,25)(14,21)(15,26)(17,20)(19,30)(23,29)(27,32)(28,31);(1,4,5,19,8,23,26,22)(2,10,11,29,13,30,24,18)(3,14,15,32,16,17,12,31)(6,27,25,21,7,28,9,20)Polycyclic presentation:
2^8|c^2=d,d^2=h,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fgh,g^a=gh,c^b=cf,d^b=dh,e^b=egh,f^b=fg,g^b=gh,e^c=egh,f^c=fgh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125729, #125730, #125731, … |
| maximal normal subgroup of | #125729, #125730, #125731, … |
| maximal quotient of | #144892, #144893, #144894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6404);