Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907869 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(17,18,22,26)(19,28,30,24)(20,32,33,47)(21,27,23,37)(25,34,41,35)(29,43,45,46)(31,40,36,39)(38,44,42,48);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,24,26,28)(20,29,33,45)(21,36,23,31)(25,38,41,42)(27,40,37,39)(32,46,47,43)(34,48,35,44);(3,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,35,22,33,21,34)(18,25,27,32,26,41,37,47)(19,29,31,44,30,45,36,48)(24,38,40,46,28,42,39,43);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(18,26)(19,30)(20,33)(27,37)(31,36)(34,35)(38,42)(43,46);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,27,26,37)(19,31,30,36)(20,34,33,35)(24,39,28,40)(25,32,41,47)(29,44,45,48)(38,43,42,46)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6817328);