Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
1251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(10,13)(11,15)(17,18,23,25,21,27,22,28)(19,29,36,38,30,24,32,40)(20,26,37,41,34,44,35,43)(31,42,46,47,33,39,45,48);(1,2,5,6)(3,7,4,8)(10,13)(11,15)(17,19,21,30)(18,24,27,29)(20,33,34,31)(22,32,23,36)(25,40,28,38)(26,42,44,39)(35,46,37,45)(41,47,43,48);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(18,25)(19,30)(20,34)(22,23)(24,38)(26,43)(27,28)(29,40)(39,48)(41,44)(42,47)(45,46);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(17,20,21,34)(18,26,27,44)(19,31,30,33)(22,35,23,37)(24,39,29,42)(25,41,28,43)(32,45,36,46)(38,47,40,48);(9,11,12,15)(10,14,13,16)(18,27)(19,30)(20,34)(25,28)(32,36)(35,37)(39,42)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7079444);