Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #35737 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1717 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,27)(22,28)(23,26)(29,38)(30,41)(31,39)(32,40)(33,42)(34,45)(35,43)(36,44)(37,46)(47,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,29)(22,37)(23,30)(25,42)(26,41)(27,38)(28,46)(31,36)(32,35)(34,47)(39,44)(40,43)(45,48);(10,14)(13,16)(17,20)(18,25)(19,29)(21,35)(22,34)(23,36)(24,38)(26,44)(27,43)(28,45)(30,47)(31,32)(33,37)(39,40)(41,48)(42,46);(1,3,4,7)(2,6,5,8)(18,26)(19,30)(20,34)(21,36)(22,35)(25,27)(29,33)(31,47)(38,39)(40,46)(42,48)(44,45);(17,21,23,34)(18,27,26,45)(19,31,30,33)(20,35,36,22)(24,39,41,42)(25,43,44,28)(29,32,47,37)(38,40,48,46)Polycyclic presentation:
2^9|b^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486884);