Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17109 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,5,12,6,11)(3,8,16,24,15,23,14,27)(4,17,19,9,20,28,21,26)(13,29,31,22,18,32,30,25);(1,3)(2,8)(4,18)(5,15)(6,16)(7,14)(9,25)(10,27)(11,24)(12,23)(13,20)(17,29)(19,31)(21,30)(22,26)(28,32);(1,4,7,19,5,20,6,21)(2,9,10,28,12,26,11,17)(3,13,16,31,15,18,14,30)(8,22,24,32,23,25,27,29);(2,10)(3,14)(6,7)(8,23)(9,17)(11,12)(13,30)(15,16)(18,31)(19,21)(26,28)(29,32)Polycyclic presentation:
2^9|a^2=ef,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384546);