Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458569 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16320 |
| 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 | |
991 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)(3,6)(4,9)(5,19,22,14)(7,11)(8,12)(10,21,28,23)(13,17)(15,18)(16,30,31,27)(20,29,32,25)(24,26);(1,3,7,12,8,11,6,2)(4,15,24,26,18,9,17,13)(5,14,21,28,22,19,23,10)(16,29,20,30,31,25,32,27);(1,4)(2,9)(3,13)(5,20,22,32)(6,17)(7,24)(8,18)(10,27,28,30)(11,26)(12,15)(14,29,19,25)(16,21,31,23);(1,5)(2,10)(3,14)(4,16)(6,23)(7,21)(8,22)(9,25)(11,19)(12,28)(13,27)(15,29)(17,32)(18,31)(20,24)(26,30)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72338620, #72338622, #72338629 |
| maximal normal subgroup of | #72338620, #72338622, #72338629 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365766);