Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C2×#472 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,4,3,6,8)(10,14)(11,17)(12,20)(16,22)(18,19)(21,24);(9,10,12,14,13,16,20,22)(11,15,18,21,19,23,17,24);(2,3)(7,8)(9,11)(10,15)(12,18)(13,19)(14,21)(16,23)(17,20)(22,24);(1,3,5,8,4,2,6,7)(10,16)(14,22)(15,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,d^c=di,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319890, #72319899, #72319984, … |
| maximal normal subgroup of | #72319890, #72319899, #72319984, … |
| maximal quotient of | C2×#211045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118241);