Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1504 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)(17,18,19,27,22,26,30,23)(20,31,32,41,34,39,28,24)(21,35,29,38,37,25,42,44)(33,48,46,40,36,47,45,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,37,42)(24,39,41,31)(25,38,35,44)(33,46,36,45)(40,48,43,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,34)(18,24,26,41)(19,28,30,32)(21,36,37,33)(23,31,27,39)(25,43,35,40)(29,46,42,45)(38,48,44,47);(3,8)(5,7)(11,16)(13,15)(17,21)(18,25)(19,29)(20,33)(22,37)(23,38)(24,40)(26,35)(27,44)(28,45)(30,42)(31,47)(32,46)(34,36)(39,48)(41,43);(10,14)(13,15)(18,26)(21,37)(23,27)(24,41)(29,42)(31,39)(33,36)(45,46)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910247);