Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,26,31,27,32,40,42)(23,33,35,30,36,46,25,28)(24,29,39,45,38,44,50,51)(34,47,49,43,48,52,37,41);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,36)(22,28,32,30)(24,34,38,48)(25,40,35,26)(29,41,44,43)(31,46,42,33)(37,50,49,39)(45,52,51,47);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,29)(23,34)(25,37)(26,39)(27,38)(28,41)(30,43)(31,45)(32,44)(33,47)(35,49)(36,48)(40,50)(42,51)(46,52);(8,16)(11,20)(13,15)(17,19)(21,25,26,23,27,35,40,36)(22,30,31,46,32,28,42,33)(24,37,39,34,38,49,50,48)(29,43,45,52,44,41,51,47);(6,12)(7,14)(11,20)(13,15)(24,38)(29,44)(34,48)(37,49)(39,50)(41,43)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=h,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011911);