Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | |
1195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,7,8,6)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,32)(21,37)(22,27)(23,39)(24,28)(26,35)(29,43)(30,41)(31,42)(33,40)(34,45)(36,48)(38,44)(46,47);(2,5)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19,24,31)(18,25,28,42)(20,33,36,38)(21,22,30,23)(26,43,45,47)(27,41,39,37)(29,34,46,35)(32,40,48,44);(9,12,13,10)(11,15,16,14)(17,20,22,34,24,36,23,35)(18,26,39,48,28,45,27,32)(19,29,21,38,31,46,30,33)(25,40,41,47,42,44,37,43);(11,16)(14,15)(17,21,23,19,24,30,22,31)(18,25,27,37,28,42,39,41)(20,33,35,46,36,38,34,29)(26,44,32,47,45,40,48,43);(1,3,4,8)(2,6,5,7)(17,22,24,23)(18,27,28,39)(19,30,31,21)(20,34,36,35)(25,41,42,37)(26,32,45,48)(29,33,46,38)(40,47,44,43)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,c^2=hi,d^2=g,e^2=fi,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^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511292);