Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354359 |
| 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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(10,14)(12,15)(16,20)(17,21)(19,22)(23,24);(9,10,15,20,13,16,12,14)(11,17,22,24,18,23,19,21);(1,2,5,7)(3,6,4,8)(10,16)(14,20)(17,23)(21,24);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,19)(13,18)(14,21)(15,22)(16,23)(20,24);(11,18)(17,23)(19,22)(21,24)Polycyclic presentation:
2^9|b^2=fh,c^2=i,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,c^b=ch,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261586);