Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16298 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,13,7,11,6,12)(3,14,16,10,17,8,22,28)(4,18,20,9,21,29,23,26)(15,25,31,30,19,27,32,24);(1,3)(2,8)(4,19)(5,22)(6,16)(7,17)(9,24)(10,13)(11,14)(12,28)(15,21)(18,25)(20,31)(23,32)(26,30)(27,29);(1,4,5,20,7,21,6,23)(2,9,12,18,11,26,13,29)(3,15,16,31,17,19,22,32)(8,24,10,27,14,30,28,25);(1,3)(2,10)(4,15)(5,16)(6,22)(7,17)(8,13)(9,25)(11,28)(12,14)(18,24)(19,21)(20,31)(23,32)(26,27)(29,30)Polycyclic presentation:
2^9|a^2=ef,c^2=f,f^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,f^b=fi,d^c=de,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364815);