Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | #25761 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1586 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)(11,14)(12,13)(15,16)(17,18)(19,24)(20,26)(21,27)(22,25)(23,28)(29,43)(30,37)(31,38)(32,39)(33,41)(34,42)(35,40)(36,44)(45,47)(46,48);(1,3,4,8)(2,5,6,7)(17,19,22,32)(18,24,25,39)(20,30,23,29)(21,34,35,45)(26,37,28,43)(27,42,40,47)(31,36,46,33)(38,44,48,41);(10,13)(14,16)(18,25)(19,29)(20,23)(24,37)(27,40)(30,32)(31,34)(33,36)(38,47)(39,43)(42,48)(45,46);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,23)(18,26,25,28)(19,30,32,29)(21,33,35,36)(24,37,39,43)(27,41,40,44)(31,45,46,34)(38,47,48,42);(3,8)(5,7)(17,21,22,35)(18,27,25,40)(19,31,32,46)(20,33,23,36)(24,38,39,48)(26,41,28,44)(29,34,30,45)(37,47,43,42)Polycyclic presentation:
2^9|b^2=fi,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6603241);