Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #547 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
287 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,8,5,7)(4,9,12,16)(11,15,13,14)(17,18)(19,26)(20,24)(21,29)(22,25)(23,28)(27,32)(30,31);(2,7)(3,5)(4,11)(8,10)(9,16)(12,13)(19,27)(23,31)(26,32)(28,30);(1,3,6,5)(2,8,10,7)(4,11,12,13)(9,15,16,14)(17,19)(18,23)(20,28)(21,30)(22,27)(24,32)(25,31)(26,29);(1,4,6,12)(2,9,10,16)(3,11,5,13)(7,14,8,15)(17,20,22,21)(18,24,25,29)(19,28,27,30)(23,32,31,26)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cgh,d^b=de,e^b=eg,f^c=fh,e^d=eg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20235, #6359592, #6359650, … |
| maximal normal subgroup of | C2×#20235, #6359592, #6359650, … |
| maximal quotient of | #220892, #220987, #222407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13223);