Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #237527 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60700 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23);(1,2,5,7)(3,8,4,6)(9,11,14,22)(10,15,18,24)(12,21,13,20)(16,19,17,23);(1,2)(3,6)(4,8)(5,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23);(2,7)(6,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=dhi,e^c=ei,f^c=fhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144737);