Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448933 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15926 |
| 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 | |
697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,4,7,12,6,11)(3,13,16,9,17,18,25,14)(5,20,24,10,23,19,22,28)(15,26,21,32,31,29,30,27);(1,3,8,16,7,17,6,25)(2,9,4,18,12,14,11,13)(5,21,24,31,23,30,22,15)(10,27,19,26,28,32,20,29);(1,4,6,12,7,11,8,2)(3,14,16,18,17,9,25,13)(5,19,22,28,23,20,24,10)(15,29,21,32,31,26,30,27);(1,5)(2,10)(3,15,17,31)(4,19)(6,22)(7,23)(8,24)(9,26,14,29)(11,20)(12,28)(13,27,18,32)(16,21,25,30)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=e,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356129);