Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354705 |
| 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 | |
1850 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)(3,4)(5,8)(6,7)(10,14)(11,18)(12,15)(17,22)(19,20)(21,24);(1,3,4,2,6,8,5,7)(9,10,15,22,13,17,12,14)(11,16,18,24,20,23,19,21);(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,23)(22,24);(10,17)(14,22)(16,23)(21,24);(1,4,6,5)(2,3,7,8)(10,17)(14,22)(16,23)(21,24)Polycyclic presentation:
2^9|b^2=fh,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72306853, #72306854, #72306906 |
| maximal normal subgroup of | #72306853, #72306854, #72306906 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261949);