Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 355 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) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10,15,16,14,18,13,17)(11,12,20,23,21,24,19,22);(1,3,5,8)(2,6,7,4)(9,11)(10,12)(13,19)(14,21)(15,20)(16,23)(17,22)(18,24);(10,16)(11,19)(13,15)(17,18)(20,21)(22,23);(1,4,5,6)(2,3,7,8)(9,12)(10,11)(13,23)(14,24)(15,22)(16,19)(17,20)(18,21)Polycyclic presentation:
2^8|a^2=efh,b^2=e,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125731, #125945, #126349, … |
| maximal normal subgroup of | #125731, #125945, #126349, … |
| maximal quotient of | #345673, #345675, #345678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24246);