Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#385 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,14,17,13,16,22,23)(11,18,20,24,19,12,15,21);(1,2)(3,8)(4,6)(5,7)(9,11,14,20,13,19,22,15)(10,12,17,21,16,18,23,24);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,17)(13,18)(14,21)(16,20)(19,23)(22,24);(11,19)(12,18)(15,20)(21,24)Polycyclic presentation:
2^9|a^2=g,b^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=di,e^b=eh,d^c=di,f^c=fh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72214180, #72214578, #72215706, … |
| maximal normal subgroup of | #72214180, #72215706, #72215940 |
| maximal quotient of | #72214091, #72215245 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99042);