Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,23,21,19,20,24);(1,3)(2,5)(4,8)(6,7)(17,19,22,24,21,18,20,23);(10,13)(14,16)(18,19)(23,24);(3,8)(5,7);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,23,19,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=df,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72315426, #72315491 |
| maximal normal subgroup of | #72315426, #72315491 |
| maximal quotient of | #72316040 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6298318);