Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
487 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 487 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(2,3)(9,10)(11,18)(12,16)(13,15)(14,19)(17,21)(20,23)(22,24);(1,2,4,3)(5,6,7,8)(9,11)(10,14)(12,19)(13,20)(15,22)(16,18)(17,23)(21,24);(6,8)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24);(1,3,4,2)(10,16)(11,19)(15,17)(20,24)Polycyclic presentation:
2^8|b^2=egh,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123173, #123183, C2×#28262, … |
| maximal normal subgroup of | #123173, #123183, C2×#28262, … |
| maximal quotient of | #317932, #317933, #317934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21250);