Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1893 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 73 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)(4,17,20,28)(5,15)(6,10)(7,14)(8,11)(9,19,27,25)(13,18,26,21)(16,22)(23,24)(29,30,32,31);(1,3,12,2,7,15,23,10,8,16,22,11,6,24,5,14)(4,13,29,9,19,31,18,28,20,26,32,27,25,30,21,17);(1,4)(2,9)(3,13)(5,21)(6,25)(7,19)(8,20)(10,28)(11,27)(12,29)(14,17)(15,31)(16,26)(18,23)(22,32)(24,30)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^b=eh,d^c=dh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13068, #494651, #495899, … |
| maximal normal subgroup of | C2×#13068, #494651, #495899, … |
| maximal quotient of | #137615, #137655, #137980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6056);