Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1453 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,30)(20,26)(21,35)(22,28)(23,29)(24,27)(25,32)(31,45)(33,42)(34,41)(36,47)(37,48)(38,44)(39,43)(40,46);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,21)(22,37)(23,32)(24,34)(26,27)(28,42)(29,30)(31,36)(33,38)(35,41)(39,46)(40,47)(43,45)(44,48);(2,6)(5,8)(11,16)(13,15)(17,20,21,19,23,34,24,32)(18,26,35,30,29,41,27,25)(22,33,36,31,38,37,39,46)(28,42,47,45,44,48,43,40);(10,14)(13,15)(17,21,23,24)(18,27,29,35)(19,20,32,34)(22,36,38,39)(25,26,30,41)(28,43,44,47)(31,33,46,37)(40,42,45,48);(17,22,24,39,23,38,21,36)(18,28,27,43,29,44,35,47)(19,31,20,33,32,46,34,37)(25,40,41,48,30,45,26,42)Polycyclic presentation:
2^9|b^2=g,c^2=fh,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6720683);