Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
448 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 448 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:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,20)(12,15)(13,17)(14,19)(16,23)(18,21)(22,24);(1,3,4,8)(2,5,6,7)(9,11,21,23,13,12,14,22)(10,15,18,24,17,20,19,16);(9,12,13,11)(10,16,17,24)(14,22,21,23)(15,18,20,19);(2,6)(5,7)(10,17)(15,20)(16,24)(18,19)Polycyclic presentation:
2^8|b^2=eg,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #301966, #301968, #301969, … |
| maximal normal subgroup of | #301966, #301968, #301969, … |
| maximal quotient of | #316208, #316210, #316216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21123);