Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #35757 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2250 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,34)(21,39)(22,41)(23,29)(24,27)(25,33)(26,37)(28,31)(32,43)(35,45)(36,46)(38,47)(40,48)(42,44);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,35)(21,32)(22,36)(23,33)(24,42)(26,45)(27,43)(28,46)(29,30)(31,38)(34,48)(37,40)(39,44)(41,47);(10,14)(13,16)(17,20,23,37)(18,26,29,34)(19,31,33,22)(21,38,24,36)(25,28,30,41)(27,47,39,46)(32,40,42,35)(43,45,44,48);(2,6)(5,7)(17,21,23,24)(18,27,29,39)(19,32,33,42)(20,36,37,38)(22,40,31,35)(25,43,30,44)(26,46,34,47)(28,48,41,45);(3,8)(5,7)(17,22,23,31)(18,28,29,41)(19,20,33,37)(21,40,24,35)(25,34,30,26)(27,48,39,45)(32,36,42,38)(43,47,44,46)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6585487);