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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,28)(19,30,33,24)(20,26,37,46)(21,29,23,27)(25,38,44,36)(31,45,35,40)(32,41,34,42)(39,47,43,48);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,24,28,30)(20,35,37,31)(21,32,23,34)(25,43,44,39)(26,45,46,40)(27,41,29,42)(36,48,38,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,37)(18,25,28,44)(19,31,33,35)(21,36,23,38)(24,39,30,43)(26,29,46,27)(32,47,34,48)(40,42,45,41);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,31)(21,38)(22,37)(23,36)(24,40)(25,27)(28,46)(29,44)(30,45)(32,48)(33,35)(34,47)(39,41)(42,43);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,32,33,34)(20,36,37,38)(24,41,30,42)(25,26,44,46)(31,47,35,48)(39,40,43,45)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6909393);