Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
194 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 194 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) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(6,12)(7,17)(8,9)(10,22)(13,29)(14,15)(16,28)(18,19)(20,36)(21,25)(23,24)(26,35)(27,31)(32,33);(1,3,4,2)(5,6)(7,13)(8,21)(9,12)(10,25)(11,16)(14,23)(15,22)(17,30)(18,27)(19,29)(20,33)(24,28)(26,32)(31,35)(34,36);(5,7,11,20)(6,13,16,33)(8,17,23,36)(9,18,24,35)(10,26,22,19)(12,27,28,31)(14,34,21,30)(15,29,25,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=gh,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,d^b=dg,e^b=eg,g^b=gh,d^c=dg,e^c=egh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12941, #494299, #494301, … |
| maximal normal subgroup of | C2×#12941, #494299, #494301, … |
| maximal quotient of | #135886, #135924, #136148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5929);