Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(10,13)(11,15)(17,18,23,25,21,27,22,28)(19,29,35,37,30,24,32,39)(20,26,36,40,33,42,34,41)(31,43,46,47,44,38,45,48);(1,3,5,8)(2,6,4,7)(10,13)(11,15)(17,19)(18,24)(20,31)(21,30)(22,32)(23,35)(25,39)(26,38)(27,29)(28,37)(33,44)(34,45)(36,46)(40,48)(41,47)(42,43);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,25)(19,30)(20,33)(22,23)(24,37)(26,41)(27,28)(29,39)(38,48)(40,42)(43,47)(45,46);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,20,21,33)(18,26,27,42)(19,31,30,44)(22,34,23,36)(24,38,29,43)(25,40,28,41)(32,45,35,46)(37,47,39,48);(9,11,12,15)(10,14,13,16)(19,30)(20,33)(24,29)(26,42)(32,35)(34,36)(37,39)(40,41)Polycyclic presentation:
2^9|a^2=fghi,b^2=fi,c^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7079852);