Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1689 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)(9,10)(11,14)(12,16)(13,15)(17,18,22,27)(19,29,33,24)(20,25,36,43)(21,26,23,28)(30,40,32,38)(31,45,47,39)(34,48,46,41)(35,42,37,44);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,34)(21,32)(22,33)(23,30)(25,41)(26,40)(27,29)(28,38)(31,35)(36,46)(37,47)(39,42)(43,48)(44,45);(9,12,13,11)(10,14,15,16)(19,30)(21,23)(24,38)(26,28)(29,40)(31,34)(32,33)(35,37)(39,41)(42,44)(45,48)(46,47);(17,20,22,36)(18,25,27,43)(19,31,33,47)(21,35,23,37)(24,39,29,45)(26,42,28,44)(30,34,32,46)(38,41,40,48);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,28)(19,32,33,30)(20,35,36,37)(24,40,29,38)(25,42,43,44)(31,46,47,34)(39,48,45,41)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354699);