Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15495 |
| 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 | |
961 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,12,6,10)(3,13,16,8,17,24,15,25)(4,18,21,9,22,27,20,28)(14,23,19,26,31,32,30,29);(1,3,5,16,7,17,6,15)(2,8,10,13,12,25,11,24)(4,19,20,14,22,30,21,31)(9,26,27,32,28,29,18,23);(1,4)(2,9)(3,14,17,31)(5,21)(6,20)(7,22)(8,23,25,32)(10,18)(11,27)(12,28)(13,29,24,26)(15,30,16,19);(2,10)(3,15)(5,6)(9,18)(11,12)(13,24)(14,30)(16,17)(19,31)(20,21)(26,29)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341215);