Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #435461 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15526 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,13,7,12,6,23)(3,14,16,19,17,9,10,4)(5,20,21,28,22,11,24,32)(15,27,26,18,30,25,29,31);(1,3,6,16,7,17,8,10)(2,9,23,4,12,14,13,19)(5,15,21,29,22,30,24,26)(11,27,32,31,20,25,28,18);(1,4,6,9,7,19,8,14)(2,10,13,3,12,16,23,17)(5,18,21,25,22,31,24,27)(11,26,28,15,20,29,32,30);(1,5)(2,11)(3,15)(4,18)(6,21)(7,22)(8,24)(9,25)(10,26)(12,20)(13,28)(14,27)(16,29)(17,30)(19,31)(23,32)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342657);