Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13074 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1893 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3058 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
245 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 245 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)(3,13)(4,9,18,25)(5,20)(6,12)(7,23)(8,11)(10,14)(15,29,27,28)(16,22)(17,31,21,32)(19,30,24,26);(1,3,14,12,7,16,22,23,6,10,13,2,8,20,5,11)(4,15,29,25,19,31,17,30,18,27,28,9,24,32,21,26);(1,4)(2,9)(3,15)(5,21)(6,18)(7,19)(8,24)(10,27)(11,26)(12,25)(13,28)(14,29)(16,31)(17,22)(20,32)(23,30)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=eh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130237, #130239, #130284, … |
| maximal normal subgroup of | #130237, #130239, #130284, … |
| maximal quotient of | #137619, #137659, #137984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6062);