Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1485 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)(3,7,4,8)(17,18,23,27)(19,28,32,43)(20,33,34,25)(21,26,22,37)(24,30,39,31)(29,44,45,42)(35,41,36,40)(38,46,48,47);(9,10)(11,14)(12,13)(15,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,37,43)(20,29,35,46,34,45,36,47)(25,38,41,42,33,48,40,44);(2,6)(7,8)(10,13)(14,16)(17,20,23,34)(18,25,27,33)(19,29,32,45)(21,36,22,35)(24,38,39,48)(26,41,37,40)(28,42,43,44)(30,47,31,46);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(20,34)(25,33)(29,45)(35,36)(38,48)(40,41)(42,44)(46,47);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,34,36)(24,28,39,43)(25,40,33,41)(29,46,45,47)(38,44,48,42)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6453901);