Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(10,15,18,17)(16,23,24,19);(1,2,4,3)(5,6)(7,8)(9,10)(11,19)(12,17)(13,15)(14,18)(16,21)(20,24)(22,23);(1,3,4,2)(6,8)(9,11,12,20,14,22,13,21)(10,16,15,23,18,24,17,19)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fh,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11291, #358309, #358310, … |
| maximal normal subgroup of | C2×#11291, #358309, #358310, … |
| maximal quotient of | #120695, #120743, #120744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4279);