Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1721 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,30)(20,26)(21,36)(22,27)(23,28)(24,29)(25,34)(31,47)(32,41)(33,46)(35,44)(37,43)(38,45)(39,42)(40,48);(2,5)(3,8)(17,19,23,34)(18,25,28,30)(20,32,21,37)(22,33,24,39)(26,43,36,41)(27,42,29,46)(31,38,48,35)(40,45,47,44);(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,26,28,36)(19,31,34,48)(22,35,24,38)(25,40,30,47)(27,44,29,45)(32,33,37,39)(41,46,43,42);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,21,23,20)(18,26,28,36)(19,32,34,37)(22,35,24,38)(25,41,30,43)(27,45,29,44)(31,33,48,39)(40,46,47,42);(17,22,23,24)(18,27,28,29)(19,33,34,39)(20,35,21,38)(25,42,30,46)(26,44,36,45)(31,37,48,32)(40,41,47,43)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7414929);