Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #459 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1879 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,6,11,7,13,24,31,8,14,4,21,23,26,5,12)(3,15,18,28,19,27,17,29,20,32,16,10,25,30,22,9);(1,3)(2,9,14,29)(4,22)(5,16)(6,17)(7,25)(8,20)(10,26,28,13)(11,27,21,30)(12,32,31,15)(18,24)(19,23);(1,3)(2,10)(4,22)(5,16)(6,17)(7,25)(8,20)(9,26)(11,15)(12,27)(13,29)(14,28)(18,24)(19,23)(21,32)(30,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=eg,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13027, #494539, #494999, … |
| maximal normal subgroup of | C2×#13027, #494539, #494999, … |
| maximal quotient of | #136697, #136698, #136731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6015);