Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1348 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)(4,6)(5,8)(17,18,24,30)(19,31,34,45)(20,37,39,43)(21,27,32,42)(22,29,23,28)(25,36,44,35)(26,33,46,38)(40,48,41,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,23,36,24,34,22,35)(18,25,29,45,30,44,28,31)(20,21,38,41,39,32,33,40)(26,27,37,48,46,42,43,47);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,33,24,39,23,38)(18,26,28,43,30,46,29,37)(19,32,35,41,34,21,36,40)(25,42,31,48,44,27,45,47);(9,11,12,16)(10,14,13,15)(17,21,24,32)(18,27,30,42)(19,33,34,38)(20,35,39,36)(22,40,23,41)(25,43,44,37)(26,31,46,45)(28,47,29,48);(1,4,5,3)(2,6,8,7)(19,34)(21,32)(25,44)(27,42)(31,45)(35,36)(40,41)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372564);