Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,8)(5,7)(17,18,22,27,23,28,20,25)(19,29,30,38,32,42,36,24)(21,26,35,41,34,40,33,39)(31,43,44,47,45,48,46,37);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,29)(26,37)(27,42)(28,38)(33,44)(34,45)(35,46)(39,43)(40,47)(41,48);(2,7)(6,8)(10,13)(11,15)(17,20,23,22)(18,25,28,27)(19,30,32,36)(21,33,34,35)(24,29,38,42)(26,39,40,41)(31,44,45,46)(37,43,47,48);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,31)(20,33)(22,35)(23,34)(24,37)(25,39)(27,41)(28,40)(29,43)(30,44)(32,45)(36,46)(38,47)(42,48);(1,4,5,3)(2,8,7,6)(10,13)(14,16)(21,34)(26,40)(31,45)(33,35)(37,47)(39,41)(43,48)(44,46)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6431720);