Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9946 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
305 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 63 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 — 3 generators on 24 points:
(2,6)(3,7)(4,8)(9,10)(11,19)(12,15)(13,17)(14,18)(16,20)(21,23)(22,24);(1,2,5,3)(4,7,8,6)(10,15)(12,13)(16,19)(17,18)(20,22)(23,24);(1,3,5,2)(4,6,8,7)(9,11,14,21)(10,16,18,24)(12,20,13,22)(15,23,17,19)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9951, #235044, #235045, … |
| maximal normal subgroup of | C2×#9951, #235044, #235045, … |
| maximal quotient of | #102797, #102812, #102825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2939);