Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470867 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1480 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)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,29,38)(20,31,32,45)(21,26,22,36)(25,33,41,34)(28,42,44,47)(30,39,35,40)(37,46,48,43);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,27,38)(20,28,32,44)(21,35,22,30)(25,37,41,48)(26,40,36,39)(31,42,45,47)(33,46,34,43);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,22,34,23,32,21,33)(18,25,26,31,27,41,36,45)(19,28,30,43,29,44,35,46)(24,37,40,42,38,48,39,47);(1,4,5,3)(2,8,6,7)(11,15)(13,16)(19,29)(20,32)(24,38)(25,41)(30,35)(31,45)(33,34)(39,40);(10,14)(13,16)(17,21,23,22)(18,26,27,36)(19,30,29,35)(20,33,32,34)(24,39,38,40)(25,31,41,45)(28,43,44,46)(37,47,48,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6381088);