Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| 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 | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,22,25)(19,26,24,21)(20,28,29,27)(23,31,32,30);(2,6)(7,14)(8,13)(9,15)(17,19,21,18,22,24,26,25)(20,27,30,23,29,28,31,32);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,20,21,30,22,29,26,31)(18,23,24,28,25,32,19,27);(20,29)(23,32)(27,28)(30,31)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=f,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #219564, #219586, #219602, … |
| maximal normal subgroup of | #219564, #219586, #219602, … |
| maximal quotient of | #191519, #206135, #206137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12647);