Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462138 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16419 |
| 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 | |
709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,11)(3,14,16,20,18,4,17,9)(5,21,24,10,25,28,23,27)(15,30,26,19,32,22,31,29);(1,3,7,17,8,18,6,16)(2,9,11,14,13,20,12,4)(5,15,24,31,25,32,23,26)(10,22,21,29,27,30,28,19);(1,4,6,20,8,14,7,9)(2,3,12,16,13,18,11,17)(5,22,23,19,25,30,24,29)(10,26,28,32,27,31,21,15);(1,5,6,23,8,25,7,24)(2,10,11,21,13,27,12,28)(3,15,16,26,18,32,17,31)(4,19,9,22,14,29,20,30)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=fi,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369335);