Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
489 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 489 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:
(2,9)(3,14)(4,16)(5,19)(10,25)(11,27)(12,20)(13,26)(15,18)(17,30)(21,22)(23,32)(24,28)(29,31);(1,2)(3,10)(4,17)(5,20)(6,23)(7,12)(8,13)(9,19)(11,14)(15,30)(16,28)(18,25)(21,26)(22,32)(24,29)(27,31);(1,3,8,4)(2,10,13,17)(5,18,22,15)(6,14,7,16)(9,24,26,27)(11,12,28,23)(19,29,21,31)(20,25,32,30);(1,4,8,3)(2,11,13,28)(5,18,22,15)(6,16,7,14)(9,25,26,30)(10,12,17,23)(19,29,21,31)(20,24,32,27)Polycyclic presentation:
2^8|c^2=h,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dgh,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 | #486122, #486126, #486134, … |
| maximal normal subgroup of | #486122, #486126, #486134, … |
| maximal quotient of | #169657, #169658, #169721, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11388);