Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1425 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)(4,7)(5,8)(10,13)(11,15)(17,18,21,27,22,26,34,23)(19,28,31,36,30,24,38,40)(20,32,33,37,35,25,41,42)(29,39,44,48,45,43,47,46);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,16,15,14)(18,23)(20,33)(21,34)(24,36)(26,27)(28,40)(29,44)(31,38)(35,41)(37,42)(45,47)(46,48);(1,3,5,4)(2,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,28)(20,29,35,45)(21,31,34,38)(23,36,27,40)(25,39,32,43)(33,44,41,47)(37,46,42,48);(1,3,5,4)(2,6,8,7)(10,13)(14,16)(17,20,34,41,22,35,21,33)(18,25,23,37,26,32,27,42)(19,29,38,47,30,45,31,44)(24,39,36,46,28,43,40,48);(1,4,5,3)(2,7,8,6)(9,11,12,15)(10,14,13,16)(18,26)(19,30)(20,35)(23,27)(31,38)(33,41)(39,43)(46,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=gh,d^2=fg,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7030760);