Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354266 |
| 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 | |
1859 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,23)(13,18)(14,21)(15,19)(16,17)(22,24);(9,11,14,17,15,22,13,23)(10,16,18,20,19,12,21,24);(2,5)(3,8)(6,7)(9,12)(10,17)(11,21)(13,20)(14,24)(15,16)(18,22)(19,23);(2,6)(5,7)(9,13,15,14)(10,18,19,21)(11,17,22,23)(12,20,16,24);(11,22)(12,16)(17,23)(20,24)Polycyclic presentation:
2^9|b^2=f,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,d^b=dh,e^b=eh,e^c=eh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306568, #72306584, #72306716, … |
| maximal normal subgroup of | #72306568, #72306584, #72306716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261464);