Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| prime radical | #62814 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,21)(14,18)(15,19)(20,23)(22,24);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,21)(17,23)(18,24);(1,3,5,4)(2,7,8,6)(9,11)(10,16)(12,19)(13,22)(14,20)(15,17)(18,23)(21,24);(1,4,5,3)(2,7,8,6)(9,11)(10,16)(12,19)(13,22)(14,20)(15,17)(18,23)(21,24);(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,23,24,16)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,f^c=fi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213590, #72219560, #72252012, … |
| maximal normal subgroup of | #72213590, #72219560, #72252012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360128);