Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #496611 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21888 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60716 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,8,13,7,26)(3,14,19,9,20,28,17,29)(4,21,23,10,18,30,22,31)(6,25,24,27,15,32,16,12);(1,3,7,19,8,20,5,17)(2,9,11,14,13,29,26,28)(4,16,23,15,18,24,22,6)(10,27,21,32,31,12,30,25);(1,4)(2,9)(3,15)(5,23)(6,20)(7,22)(8,18)(10,27)(11,28)(12,31)(13,29)(14,26)(16,19)(17,24)(21,25)(30,32);(1,4)(2,10,13,31)(3,16)(5,22)(6,17)(7,23)(8,18)(9,27,29,12)(11,21,26,30)(14,32,28,25)(15,19)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=eh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^b=gi,e^c=ei,f^c=fh,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403810);