Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13215 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1961 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3155 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 65 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)(3,14)(4,19,23,28)(5,17)(6,11)(7,13)(8,12)(9,22,27,25)(10,15)(16,21,26,31)(18,24)(20,29,32,30);(1,3,15,13,6,18,24,11,7,10,14,2,8,17,5,12)(4,20,29,28,22,26,21,27,23,32,30,19,25,16,31,9);(1,4,7,23)(2,9)(3,16)(5,21,24,31)(6,22,8,25)(10,26)(11,19)(12,28)(13,27)(14,29,15,30)(17,32)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=cg,e^b=eh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13215, #495144, #495573, … |
| maximal normal subgroup of | C2×#13215, #495144, #495573, … |
| maximal quotient of | #140556, #140558, #140598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6203);