Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24061 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60710 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2873 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
775 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)(3,11)(4,16)(5,21)(6,9)(7,14)(8,20)(10,19)(12,29)(13,27)(15,24)(17,23)(18,31)(22,28)(25,32)(26,30);(1,2)(3,12,13,7)(4,8)(5,10)(6,9)(11,14,27,29)(15,26,22,23)(16,20)(17,28,30,24)(18,25)(19,21)(31,32);(1,3,5,22,6,13,19,15)(2,7,10,26,9,12,21,23)(4,17,20,29,18,30,32,14)(8,24,16,27,25,28,31,11);(4,18)(8,25)(11,27)(14,29)(16,31)(17,30)(20,32)(24,28)Polycyclic presentation:
2^9|b^2=e,c^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fhi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72214675 |
| maximal normal subgroup of | #72214675 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407504);