Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
360 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 360 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:
(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18)(19,24)(20,29)(21,31)(22,26)(23,28)(25,30)(27,32);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,21,28,22,27,30,20)(18,23,25,24,26,29,31,32);(1,3,5,9)(2,7,8,14)(4,10,11,15)(6,12,13,16)(17,20,30,27,22,28,21,19)(18,24,31,23,26,32,25,29);(3,9)(7,14)(10,15)(12,16)(19,27)(20,28)(23,29)(24,32)Polycyclic presentation:
2^8|b^2=e,c^2=fg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dh,d^c=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | #143168, C2×#19717, #6357363, … |
| maximal normal subgroup of | #143168, C2×#19717, #6357363, … |
| maximal quotient of | #191523, #206261, #206268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12705);