Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45092 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,23,29)(19,27,32,25)(20,34,38,46)(21,40,33,42)(22,30,24,28)(26,39,45,37)(31,47,48,43)(35,44,36,41);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,25,29,27)(20,35,38,36)(21,24,33,22)(26,43,45,47)(28,40,30,42)(31,39,48,37)(34,41,46,44);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,24,39,23,38,22,37)(18,26,30,46,29,45,28,34)(19,31,21,36,32,48,33,35)(25,41,40,47,27,44,42,43);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,24,32,23,33,22,19)(18,27,30,42,29,25,28,40)(20,36,39,48,38,35,37,31)(26,44,46,43,45,41,34,47);(2,6)(7,8)(10,14)(13,15)(17,22,23,24)(18,28,29,30)(19,21,32,33)(20,37,38,39)(25,40,27,42)(26,34,45,46)(31,36,48,35)(41,47,44,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=fg,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7499431);