Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #392 |
| prime radical | C23 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
238 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,23,20)(19,27,31,28)(21,26,22,25)(24,30,32,29);(6,15)(9,16)(17,19,23,31)(18,24,20,32)(21,30,22,29)(25,28,26,27);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,20,23,18)(19,28,31,27)(21,25,22,26)(24,29,32,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dg,f^b=fg,d^c=dh,e^c=egh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #93232, #93236, #125976, … |
| maximal normal subgroup of | #93232, #93236, #125976, … |
| maximal quotient of | #102434, #102439, #102448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3025);