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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13)(17,18,22,27)(19,29,32,24)(20,33,37,25)(21,26,23,28)(30,40,35,43)(31,45,38,41)(34,42,47,48)(36,46,39,44);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,27,29)(20,34,37,47)(21,31,23,38)(25,42,33,48)(26,41,28,45)(30,39,35,36)(40,46,43,44);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,24)(20,35)(21,38)(22,32)(23,31)(25,43)(26,45)(27,29)(28,41)(30,37)(33,40)(34,39)(36,47)(42,46)(44,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,39)(24,40)(26,44)(27,33)(28,46)(29,43)(31,47)(32,35)(34,38)(41,48)(42,45);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,31,32,38)(20,36,37,39)(24,41,29,45)(25,44,33,46)(30,47,35,34)(40,48,43,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7497017);