Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,19)(12,15)(13,18)(14,23)(16,21)(17,20)(22,24);(1,2,4,5)(3,6,7,8)(9,11,15,22)(10,16,12,17)(13,21,14,20)(18,24,23,19);(2,5)(3,7)(9,11)(10,17)(12,16)(13,20)(14,21)(15,22)(18,24)(19,23);(1,3)(2,6)(4,7)(5,8)(9,12,15,10)(11,17,22,16)(13,18,14,23)(19,21,24,20);(2,5)(3,7)(9,13,15,14)(10,18,12,23)(11,20,22,21)(16,19,17,24)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503055);