Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460301 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16361 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,12,22,28)(3,8,17,25,15,26,16,24)(4,10,19,29,5,20,21,9)(13,30,18,23,31,27,14,32);(1,3)(2,8)(4,18)(5,14)(6,15)(7,16)(9,27)(10,23)(11,24)(12,26)(13,19)(17,22)(20,32)(21,31)(25,28)(29,30);(1,4)(2,9)(3,13)(5,6)(7,21)(8,23)(10,28)(11,20)(12,29)(14,17)(15,31)(16,18)(19,22)(24,27)(25,30)(26,32);(1,5,6,4)(2,10)(3,14,15,18)(7,21,22,19)(8,23)(9,28)(11,29)(12,20)(13,16,31,17)(24,30)(25,27)(26,32)Polycyclic presentation:
2^9|a^2=g,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367497);