Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | #37656 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,26)(22,37)(23,27)(25,32)(28,40)(29,38)(31,46)(33,47)(34,48)(35,42)(36,43)(39,45)(41,44);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,29,36,35)(25,30,40,47)(26,38,43,42)(31,44,34,45)(39,46,41,48);(11,16)(13,15)(17,20,22,32,23,33,19,28)(18,25,24,30,27,40,37,47)(21,34,35,44,36,31,29,45)(26,41,38,48,43,39,42,46);(2,5)(6,8)(10,14)(13,15)(17,21)(18,26)(19,29)(20,31)(22,35)(23,36)(24,38)(25,39)(27,43)(28,44)(30,46)(32,45)(33,34)(37,42)(40,41)(47,48);(1,3,4,7)(2,6,5,8)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,35,36,29)(25,30,40,47)(26,42,43,38)(31,45,34,44)(39,48,41,46)Polycyclic presentation:
2^9|b^2=i,c^2=g,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6704765);