Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484139 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
584 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)(7,8)(9,10,14,19,16,20,15,24)(11,17,21,23,22,18,13,12);(1,2)(3,7)(4,8)(5,6)(9,11,15,13,16,22,14,21)(10,17,24,12,20,18,19,23);(1,3,5,4)(2,7,6,8)(9,12,13,20,15,17,22,24,16,23,21,10,14,18,11,19);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=ef,d^2=e,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72264870, #72264898, #72264915, … |
| maximal normal subgroup of | #72264870, #72264898, #72264915, … |
| maximal quotient of | #72309679, #72309801, #72309840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405820);