Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 420 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(1,2,6,8,4,7,3,5);(9,11,12,19,14,21,13,20)(10,15,22,24,17,23,16,18);(2,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)Polycyclic presentation:
2^8|b^2=eg,c^2=fh,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #490786, #490788, #490790, … |
| maximal normal subgroup of | #490786, #490788, #490790, … |
| maximal quotient of | #191442, #206130, #206133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12635);