Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428870 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15391 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,9)(3,13,17,25,14,23,16,8)(4,18,21,10,22,26,20,28)(15,31,32,29,30,24,19,27);(1,3,5,16,7,14,6,17)(2,8,11,13,12,25,9,23)(4,19,20,30,22,32,21,15)(10,27,26,31,28,29,18,24);(2,9)(3,14)(5,6)(8,23)(10,26)(11,12)(13,25)(15,30)(18,28)(20,21)(24,27)(29,31);(1,4)(2,10,12,28)(3,15)(5,21)(6,20)(7,22)(8,24,25,31)(9,26,11,18)(13,29,23,27)(14,30)(16,32)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251788, #72251792, #72251800, … |
| maximal normal subgroup of | #72251788, #72251792, #72251800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336066);