Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18192 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
494 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 494 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:
(2,8)(3,14)(4,16)(5,19)(9,27)(10,23)(11,25)(12,20)(13,26)(17,24)(21,32)(28,29);(1,2)(3,9)(4,17)(5,20)(6,12)(7,13)(8,19)(10,16)(11,22)(14,29)(15,27)(18,24)(21,25)(23,31)(26,32)(28,30);(1,3,7,4)(2,9,13,17)(5,18,21,15)(6,14,22,16)(8,23,26,28)(10,12,29,11)(19,31,32,30)(20,24,25,27);(1,4,7,3)(2,10,13,29)(5,18,21,15)(6,16,22,14)(8,24,26,27)(9,12,17,11)(19,31,32,30)(20,23,25,28)Polycyclic presentation:
2^8|c^2=h,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #486116, #486120, #486128, … |
| maximal normal subgroup of | #486116, #486120, #486128, … |
| maximal quotient of | #167045, #167047, #167109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11204);