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 | |
1455 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)(17,18,22,25,23,27,20,32)(19,28,29,40,31,38,37,24)(21,34,36,46,35,26,33,41)(30,44,43,39,45,42,48,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,32)(33,43)(34,47)(35,45)(36,48)(41,44)(42,46);(2,7)(6,8)(10,13)(11,15)(17,20,23,22)(18,25,27,32)(19,29,31,37)(21,33,35,36)(24,38,40,28)(26,41,34,46)(30,43,45,48)(39,44,47,42);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,33)(22,36)(23,35)(24,39)(25,41)(27,34)(28,42)(29,43)(31,45)(32,46)(37,48)(38,44)(40,47);(1,4,5,3)(2,8,7,6)(10,13)(14,16)(18,27)(21,35)(24,40)(25,32)(28,38)(30,45)(33,36)(43,48)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=i,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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, 6431889);