Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13442 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2054 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(2,9)(3,15)(4,20)(5,6)(7,21)(11,24)(12,14)(13,22)(16,25)(17,31)(18,23)(19,29)(26,30)(28,32);(1,2)(3,16)(4,11)(5,12)(6,22)(7,24)(8,14)(9,20)(10,17)(13,21)(15,26)(18,32)(19,28)(23,27)(25,31)(29,30);(1,3,4,17,21,31,5,15,8,19,6,23,7,18,20,29)(2,10,9,16,24,30,22,32,14,27,12,28,13,26,11,25)Polycyclic presentation:
2^8|c^2=d,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,g^b=gh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13442, #499179, #499514, … |
| maximal normal subgroup of | C2×#13442, #499179, #499514, … |
| maximal quotient of | #145359, #145360, #145363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6430);