Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,24,39,23,28,22,27)(19,29,21,25,31,45,32,42)(20,33,40,44,36,26,37,48)(30,47,38,43,34,46,35,41);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,25,28,42)(20,34,36,30)(21,24,32,22)(26,43,33,41)(27,45,39,29)(35,40,38,37)(44,46,48,47);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20,23,36)(18,26,28,33)(19,30,31,34)(21,38,32,35)(22,37,24,40)(25,41,42,43)(27,44,39,48)(29,46,45,47);(3,8)(5,7)(11,16)(13,15)(17,21,22,19,23,32,24,31)(18,25,27,29,28,42,39,45)(20,35,37,34,36,38,40,30)(26,43,44,47,33,41,48,46);(10,14)(13,15)(19,31)(20,36)(21,32)(25,42)(26,33)(29,45)(37,40)(44,48)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=h,d^2=fgi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6986805);