Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| 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 | |
1492 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)(19,28,32,42)(20,25,33,39)(21,26,22,36)(24,30,38,31)(29,41,43,48)(34,40,35,46)(37,44,47,45);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,42)(20,29,35,45,33,43,34,44)(25,37,40,41,39,47,46,48);(1,3,5,4)(2,7,6,8)(11,16)(13,14)(17,20,23,33)(18,25,27,39)(19,29,32,43)(21,34,22,35)(24,37,38,47)(26,40,36,46)(28,41,42,48)(30,44,31,45);(1,4,5,3)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(20,33)(25,39)(29,43)(34,35)(37,47)(40,46)(41,48)(44,45);(11,16)(13,14)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,33,35)(24,28,38,42)(25,40,39,46)(29,44,43,45)(37,41,47,48)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=dhi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6522123);