Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | |
1389 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)(4,6)(5,8)(17,18)(19,28)(20,34)(21,29)(22,40)(23,27)(24,30)(25,32)(26,38)(31,45)(33,46)(35,48)(36,44)(37,43)(39,47)(41,42);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,23,32,24,33,21,22)(18,25,29,28,30,40,27,46)(20,35,36,41,38,31,39,37)(26,43,44,45,34,42,47,48);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20)(18,26)(19,31)(21,39)(22,41)(23,36)(24,38)(25,42)(27,47)(28,48)(29,44)(30,34)(32,37)(33,35)(40,43)(45,46);(2,6)(5,7)(10,14)(13,16)(17,21,24,23)(18,27,30,29)(19,32,33,22)(20,36,38,39)(25,28,40,46)(26,44,34,47)(31,41,35,37)(42,45,43,48);(2,6)(5,7)(17,22,21,33,24,32,23,19)(18,28,27,25,30,46,29,40)(20,37,39,31,38,41,36,35)(26,45,47,43,34,48,44,42)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=i,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507555);