Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| 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 | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
431 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 431 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,9,11,16)(4,12,14,8)(6,13,15,10)(17,18,22,19)(20,27,28,32)(21,26,31,24)(23,30,25,29);(1,3)(2,6)(4,13)(5,11)(7,15)(8,16)(9,12)(10,14)(17,19,22,18)(20,25,28,23)(21,24,31,26)(27,30,32,29);(4,14)(8,12)(9,16)(10,13)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,27)(29,31);(2,7)(6,15)(8,12)(9,16)Polycyclic presentation:
2^8|a^2=h,b^2=gh,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #245907, #245908, #245919, … |
| maximal normal subgroup of | #245907, #245908, #245919, … |
| maximal quotient of | #165889, #165891, #165921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11995);