Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 | |
1337 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)(9,10)(11,15)(12,16)(13,14)(17,18,24,30)(19,26,33,40)(20,25,32,41)(21,27,36,44)(22,29,23,28)(31,42,47,43)(34,39,35,48)(37,46,38,45);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,23,32,24,33,22,20)(18,25,29,40,30,41,28,26)(21,31,38,35,36,47,37,34)(27,39,46,43,44,48,45,42);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,22,33,24,32,23,19)(18,26,28,41,30,40,29,25)(21,35,37,31,36,34,38,47)(27,43,45,39,44,42,46,48);(9,11,13,12)(10,15,14,16)(17,21)(18,27)(19,31)(20,34)(22,37)(23,38)(24,36)(25,39)(26,42)(28,45)(29,46)(30,44)(32,35)(33,47)(40,43)(41,48);(1,4,5,3)(2,6,8,7)(9,12,13,11)(10,15,14,16)(21,36)(27,44)(31,47)(34,35)(37,38)(39,48)(42,43)(45,46)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,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6393086);