Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| 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 | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
438 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 438 conjugacy classes of subgroups — too many to draw. Only its 115 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)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(2,7)(6,15)(8,12)(9,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=h,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 | #246995, #246996, #246997, … |
| maximal normal subgroup of | #246995, #246996, #246997, … |
| maximal quotient of | #168559, #168616, #168666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11328);