Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,24,29)(19,25,32,42)(20,33,35,46)(21,38,31,41)(22,28,23,27)(26,37,45,36)(30,43,47,48)(34,40,39,44);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,24,32)(18,25,29,42)(20,34,35,39)(21,23,31,22)(26,43,45,48)(27,41,28,38)(30,37,47,36)(33,40,46,44);(10,13)(14,15)(17,20,23,37,24,35,22,36)(18,26,28,46,29,45,27,33)(19,30,21,39,32,47,31,34)(25,40,41,48,42,44,38,43);(9,11,12,16)(10,14,13,15)(17,21,23,32,24,31,22,19)(18,25,28,41,29,42,27,38)(20,34,37,30,35,39,36,47)(26,44,46,43,45,40,33,48);(1,3,4,8)(2,6,5,7)(20,35)(26,45)(30,47)(33,46)(34,39)(36,37)(40,44)(43,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503630);