Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6)(7,8)(9,10,15,11)(12,20,23,24)(13,18,14,17)(16,22,19,21);(1,2,4,3)(9,11,15,10)(12,20,23,24)(13,17,14,18)(16,22,19,21);(6,8)(9,12,15,23)(10,16,11,19)(13,22,14,21)(17,24,18,20);(1,3,4,2)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #244799, #244802, #244814, … |
| maximal normal subgroup of | #244799, #244802, #244814, … |
| maximal quotient of | #261274, #261285, #261294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17231);