Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355483 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,8)(3,4,7,5)(9,10,14,17)(11,13,16,23)(12,19,22,18)(15,20,24,21);(1,3,5,2,6,7,4,8)(9,11,13,17,14,16,23,10)(12,18,20,24,22,19,21,15);(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,24)(21,23);(9,13,14,23)(10,16,17,11)(12,20,22,21)(15,19,24,18);(1,4,6,5)(2,7,8,3)(10,17)(11,16)(15,24)(18,19)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283723, #72283748, #72306900, … |
| maximal normal subgroup of | #72283723, #72283748, #72306900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262739);