Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396125 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14730 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,8,6,20)(3,12,15,21,16,23,13,9)(4,10,17,25,18,22,19,26)(14,29,28,32,30,27,31,24);(2,8)(3,13)(5,6)(9,23)(10,22)(12,21)(14,28)(15,16)(17,19)(24,29)(27,32)(30,31);(1,3)(2,9)(4,14)(5,13)(6,15)(7,16)(8,21)(10,24)(11,23)(12,20)(17,31)(18,30)(19,28)(22,32)(25,27)(26,29);(1,4)(2,10,8,22)(3,14)(5,19)(6,17)(7,18)(9,24,21,32)(11,26,20,25)(12,27,23,29)(13,28)(15,31)(16,30)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303321);