Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1321 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,32)(21,35)(22,27)(23,28)(25,31)(26,37)(29,44)(30,40)(33,43)(34,41)(36,47)(38,48)(39,46)(42,45);(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,27,40)(20,33,31,23)(21,36,37,34)(25,28,32,43)(26,41,35,47)(29,45,46,38)(39,48,44,42);(2,6)(7,8)(10,14)(11,16)(17,20,23,19,22,31,33,30)(18,25,28,40,27,32,43,24)(21,34,38,29,37,36,45,46)(26,41,42,44,35,47,48,39);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(21,37)(26,35)(29,46)(34,36)(38,45)(39,44)(41,47)(42,48);(1,4,5,3)(2,8,6,7)(17,21,23,38,22,37,33,45)(18,26,28,42,27,35,43,48)(19,29,31,36,30,46,20,34)(24,39,25,41,40,44,32,47)Polycyclic presentation:
2^9|b^2=i,c^2=fh,d^2=fgi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6749020);