Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3460 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
499 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 32 points:
(1,2,7,13)(3,15,17,28)(4,11,5,21)(6,12,8,14)(9,22,26,16)(10,20,23,19)(18,25,24,32)(27,31,29,30);(1,3)(2,9)(4,16)(5,22)(6,24)(7,17)(8,18)(10,25)(11,28)(12,29)(13,26)(14,27)(15,21)(19,31)(20,30)(23,32);(1,2,8,14)(3,9,18,27)(4,10,20,21)(5,23,19,11)(6,12,7,13)(15,16,25,30)(17,26,24,29)(22,32,31,28);(1,4,7,5)(2,10,13,23)(3,16,17,22)(6,19,8,20)(9,25,26,32)(11,14,21,12)(15,29,28,27)(18,30,24,31)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #485971, #485972, #485976, … |
| maximal normal subgroup of | #485971, #485972, #485976, … |
| maximal quotient of | #168516, #168525, #168576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11319);