Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1444 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)(3,7)(4,6)(5,8)(17,18,22,27)(19,28,23,24)(20,31,34,39)(21,35,37,26)(25,32,41,29)(30,38,36,42)(33,46,47,44)(40,43,48,45);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,36,37,30)(25,39,41,31)(26,42,35,38)(33,43,47,45)(40,44,48,46);(3,8)(5,7)(10,13)(14,15)(17,20,22,34)(18,25,27,41)(19,29,23,32)(21,33,37,47)(24,31,28,39)(26,40,35,48)(30,43,36,45)(38,44,42,46);(2,6)(5,7)(17,21)(18,26)(19,30)(20,33)(22,37)(23,36)(24,38)(25,40)(27,35)(28,42)(29,43)(31,44)(32,45)(34,47)(39,46)(41,48);(9,11,12,16)(10,14,13,15)(18,27)(21,37)(24,28)(25,41)(30,36)(31,39)(33,47)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6707536);