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 | |
1457 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)(17,18,23,36,22,27,20,25)(19,28,29,24,32,38,31,40)(21,34,37,41,35,26,33,47)(30,44,43,48,45,42,46,39);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,38)(26,39)(27,40)(28,36)(33,43)(34,48)(35,45)(37,46)(41,44)(42,47);(2,7)(6,8)(10,13)(11,15)(17,20,22,23)(18,25,27,36)(19,29,32,31)(21,33,35,37)(24,38,40,28)(26,41,34,47)(30,43,45,46)(39,44,48,42);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,33)(22,35)(23,37)(24,39)(25,41)(27,34)(28,42)(29,43)(31,46)(32,45)(36,47)(38,44)(40,48);(1,4,5,3)(2,8,7,6)(10,13)(14,16)(21,35)(26,34)(30,45)(33,37)(39,48)(41,47)(42,44)(43,46)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6431832);