Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1421 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)(3,6)(4,5)(7,8)(11,15)(13,16)(17,18,23,27,22,28,21,29)(19,30,38,42,33,24,32,41)(20,34,35,25,36,26,37,43)(31,40,46,44,47,45,48,39);(2,5)(3,7)(9,10)(11,16)(12,14)(13,15)(17,19,22,33)(18,24,28,30)(20,31,36,47)(21,32,23,38)(25,39,43,44)(26,40,34,45)(27,41,29,42)(35,46,37,48);(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,31)(21,35)(22,36)(23,37)(24,39)(26,29)(27,34)(28,43)(30,44)(32,46)(33,47)(38,48)(40,42)(41,45);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20)(18,26)(19,31)(21,37)(22,36)(23,35)(24,40)(25,29)(27,43)(28,34)(30,45)(32,48)(33,47)(38,46)(39,42)(41,44);(2,5)(6,8)(10,14)(13,16)(17,21,22,23)(18,27,28,29)(19,32,33,38)(20,35,36,37)(24,41,30,42)(25,34,43,26)(31,46,47,48)(39,45,44,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7180715);