Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479149 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17188 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,14,6,13)(3,11,5,23,17,29,16,9)(4,18,21,10,22,26,20,27)(15,30,24,32,19,25,31,28);(1,3,6,16,8,17,7,5)(2,9,12,11,14,23,13,29)(4,19,20,24,22,15,21,31)(10,25,26,28,27,30,18,32);(1,4,6,20,8,22,7,21)(2,10,12,26,14,27,13,18)(3,15,16,31,17,19,5,24)(9,25,11,28,23,30,29,32);(1,5,6,17,8,16,7,3)(2,11,12,9,14,29,13,23)(4,19,20,31,22,15,21,24)(10,25,26,32,27,30,18,28)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=e,d^2=fi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,h^c=hi,e^d=ei47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386346);