Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,8,4,7)(9,10)(11,14)(12,15)(13,16)(17,18,23,27)(19,28,31,39)(20,25,33,42)(21,26,22,37)(24,36,40,30)(29,44,32,46)(34,43,35,48)(38,47,41,45);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,36,23,31,22,30)(18,24,37,39,27,40,26,28)(20,32,34,45,33,29,35,47)(25,41,48,44,42,38,43,46);(1,3,5,4)(2,7,6,8)(11,16)(13,14)(17,20,23,33)(18,25,27,42)(19,29,31,32)(21,34,22,35)(24,38,40,41)(26,43,37,48)(28,44,39,46)(30,45,36,47);(1,4,5,3)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(20,33)(25,42)(29,32)(34,35)(38,41)(43,48)(44,46)(45,47);(11,16)(13,14)(17,21,23,22)(18,26,27,37)(19,30,31,36)(20,34,33,35)(24,39,40,28)(25,43,42,48)(29,45,32,47)(38,46,41,44)Polycyclic presentation:
2^9|a^2=gi,b^2=fgh,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6522223);