Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,30,34,25)(20,27,21,26)(22,38,24,28)(31,40,36,46)(32,41,48,47)(33,42,39,43)(35,44,37,45);(1,3,5,4)(2,6,8,7)(11,16)(13,15)(17,19)(18,25)(20,31)(21,36)(22,33)(23,34)(24,39)(26,40)(27,46)(28,42)(29,30)(32,37)(35,48)(38,43)(41,44)(45,47);(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,26,29,27)(19,31,34,36)(22,37,24,35)(25,40,30,46)(28,45,38,44)(32,39,48,33)(41,42,47,43);(10,14)(13,15)(17,21,23,20)(18,27,29,26)(19,32,34,48)(22,37,24,35)(25,41,30,47)(28,45,38,44)(31,39,36,33)(40,42,46,43);(1,4,5,3)(2,7,8,6)(17,22,23,24)(18,28,29,38)(19,33,34,39)(20,35,21,37)(25,42,30,43)(26,44,27,45)(31,48,36,32)(40,41,46,47)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7441225);