Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | Aut(G) order 16384 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
465 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 465 conjugacy classes of subgroups — too many to draw. Only its 117 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(17,20,21,27)(18,23,24,30)(19,25,26,32)(22,28,29,31);(2,7)(6,15)(8,12)(9,16)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=h,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #246355, #246356, #246363, … |
| maximal normal subgroup of | #246355, #246356, #246363, … |
| maximal quotient of | #165890, #165900, #165928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11136);