Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1469 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)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,24)(21,26)(22,25)(27,34)(28,36)(29,35)(30,37)(31,38)(32,39)(33,40)(41,45)(42,46)(43,47)(44,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(20,30)(21,32)(24,37)(26,39)(27,41)(29,43)(31,33)(34,45)(35,47)(38,40)(42,44)(46,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,22,28)(18,23,25,36)(20,27,31,42)(21,29,32,43)(24,34,38,46)(26,35,39,47)(30,41,33,44)(37,45,40,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,24)(19,27)(21,33)(22,31)(23,34)(25,38)(26,40)(28,42)(29,44)(30,32)(35,48)(36,46)(37,39)(41,43)(45,47);(2,6)(7,8)(10,14)(13,15)(18,25)(19,28)(20,31)(26,39)(29,43)(30,33)(34,46)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fgi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7479182);