Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(17,18,21,25)(19,27,31,23)(20,32,33,24)(22,26,35,40)(28,38,30,36)(29,37,44,41)(34,45,46,39)(42,47,43,48);(1,3,4,8)(2,5,6,7)(17,19)(18,23)(20,29)(21,31)(22,28)(24,37)(25,27)(26,36)(30,35)(32,41)(33,44)(34,42)(38,40)(39,47)(43,46)(45,48);(2,6)(5,7)(10,14)(15,16)(19,28)(20,33)(22,35)(23,36)(24,32)(26,40)(27,38)(29,43)(30,31)(37,48)(41,47)(42,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,21,33)(18,24,25,32)(19,29,31,44)(22,34,35,46)(23,37,27,41)(26,39,40,45)(28,42,30,43)(36,47,38,48);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,30)(22,35)(23,38)(26,40)(27,36)(28,31)(29,43)(34,46)(37,48)(39,45)(41,47)(42,44)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6650217);