Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1336 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)(19,23)(20,31)(21,25)(22,27)(24,28)(26,30)(29,37)(32,44)(33,36)(34,40)(35,41)(38,42)(39,43)(45,48)(46,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,30)(18,23,27,26)(20,28,33,43)(21,34,35,29)(24,36,39,31)(25,40,41,37)(32,45,46,42)(38,44,48,47);(2,6)(5,7)(10,14)(15,16)(17,20,30,43,22,33,19,28)(18,24,23,36,27,39,26,31)(21,32,34,45,35,46,29,42)(25,38,37,47,41,48,40,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,35)(18,25,27,41)(19,29,30,34)(20,32,33,46)(23,37,26,40)(24,38,39,48)(28,42,43,45)(31,44,36,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(20,33)(21,35)(24,39)(25,41)(28,43)(29,34)(31,36)(37,40)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6784627);