Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
444 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(18,23)(20,21)(24,31)(25,28)(26,29)(30,32);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18,22,25)(19,24,27,32)(20,23,21,28)(26,30,29,31);(17,19)(18,24)(20,26)(21,29)(22,27)(23,30)(25,32)(28,31);(17,20,22,21)(18,23,25,28)(19,26,27,29)(24,30,32,31)Polycyclic presentation:
2^8|a^2=gh,b^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142450, #142451, #142454, … |
| maximal normal subgroup of | #142450, #142451, #142454, … |
| maximal quotient of | #168551, #168608, #168659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11299);