Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 | #38903 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1577 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)(3,6)(4,5)(7,8)(17,18,22,27,21,26,36,24)(19,28,31,38,30,23,37,40)(20,32,35,41,34,25,42,44)(29,39,45,48,33,43,46,47);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,33)(21,30)(22,31)(24,38)(25,43)(26,28)(27,40)(29,34)(32,39)(35,46)(36,37)(41,48)(42,45)(44,47);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(18,24)(20,34)(22,36)(23,38)(25,44)(26,27)(28,40)(29,33)(31,37)(32,41)(39,48)(43,47);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,34)(18,25,26,32)(19,29,30,33)(22,35,36,42)(23,39,28,43)(24,41,27,44)(31,45,37,46)(38,47,40,48);(10,13)(14,16)(18,26)(19,30)(20,34)(24,27)(31,37)(35,42)(39,43)(47,48)Polycyclic presentation:
2^9|a^2=g,c^2=fi,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6844169);