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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
437 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(4,5)(7,8)(10,14)(12,15)(16,20)(17,21)(19,22)(23,24);(1,2,5,8,6,7,4,3)(9,10,15,21,13,17,12,14)(11,16,22,24,18,23,19,20);(1,3,4,7,6,8,5,2)(9,11)(10,16)(12,19)(13,18)(14,20)(15,22)(17,23)(21,24);(2,7)(3,8)(11,18)(16,23)(19,22)(20,24)Polycyclic presentation:
2^8|b^2=eg,c^2=fh,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 | #139431, #153539, #153549, … |
| maximal normal subgroup of | #139431, #153539, #153549, … |
| maximal quotient of | #191525, #206264, #206272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12715);