Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1417 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,28)(20,33)(21,27)(22,24)(23,38)(25,32)(26,36)(29,45)(30,46)(31,39)(34,44)(35,43)(37,42)(40,48)(41,47);(1,3,7,4)(2,5,8,6)(9,10)(11,14)(12,13)(15,16)(18,24)(19,29)(26,33)(27,38)(30,34)(31,32)(42,43)(47,48);(1,4,7,3)(2,5,8,6)(10,13)(14,15)(17,19,21,31,22,29,23,32)(18,25,38,45,24,39,27,28)(20,34,37,47,36,30,35,48)(26,41,42,44,33,40,43,46);(1,5,7,6)(2,3,8,4)(9,11,12,16)(10,14,13,15)(17,20,22,36)(18,26,24,33)(19,30,29,34)(21,37,23,35)(25,40,39,41)(27,43,38,42)(28,44,45,46)(31,48,32,47);(1,6,7,5)(2,3,8,4)(17,21,22,23)(18,27,24,38)(19,31,29,32)(20,35,36,37)(25,28,39,45)(26,42,33,43)(30,47,34,48)(40,46,41,44)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=fg,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893144);