Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18810 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3515 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
438 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 438 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,5,8)(3,9,10,16)(4,12,14,7)(6,13,15,11)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(1,3,5,10)(2,6,8,15)(4,13,14,11)(7,16,12,9)(17,18)(19,23)(20,24)(21,28)(22,25)(26,31)(27,32)(29,30);(17,19)(18,23)(20,26)(21,29)(22,27)(24,31)(25,32)(28,30);(3,10)(6,15)(9,16)(11,13)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #273706, #273721, #273734, … |
| maximal normal subgroup of | #273706, #273721, #273734, … |
| maximal quotient of | #168936, #169000, #169046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11825);