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×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3184 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
356 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 356 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,8)(5,9)(6,16)(7,10)(12,14)(13,15)(17,18)(19,25)(20,30)(21,26)(22,31)(23,27)(24,28)(29,32);(1,3,6,8,7,13,5,12)(2,4,9,11,10,14,16,15)(17,19,22,28,23,29,21,20)(18,24,26,25,27,30,31,32);(1,4,7,14)(2,8,10,12)(3,9,13,16)(5,15,6,11)(17,20,21,29,23,28,22,19)(18,25,31,24,27,32,26,30);(1,5,7,6)(2,9,10,16)(3,12,13,8)(4,11,14,15)(17,21,23,22)(18,26,27,31)(19,28,29,20)(24,32,30,25)Polycyclic presentation:
2^8|b^2=e,c^2=fg,d^2=g,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 | #143167, C2×#19724, #6357363, … |
| maximal normal subgroup of | #143167, C2×#19724, #6357363, … |
| maximal quotient of | #191523, #206260, #206269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12712);