Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,36,22,26,21,27)(19,24,29,39,31,38,30,40)(20,32,33,25,35,42,34,41)(28,43,46,37,45,48,44,47);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,38)(27,40)(32,43)(33,46)(34,44)(35,45)(36,39)(41,47)(42,48);(10,11)(14,15)(17,20,22,35)(18,25,26,41)(19,28,31,45)(21,33,23,34)(24,37,38,47)(27,42,36,32)(29,44,30,46)(39,43,40,48);(1,4,5,3)(2,6,7,8)(12,16)(14,15)(18,26)(24,38)(25,41)(27,36)(32,42)(37,47)(39,40)(43,48);(9,12,13,16)(10,14,11,15)(17,21,22,23)(18,27,26,36)(19,29,31,30)(20,33,35,34)(24,39,38,40)(25,42,41,32)(28,44,45,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6456394);