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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18)(19,30)(20,34)(21,29)(22,27)(23,28)(24,32)(25,33)(26,35)(31,44)(36,47)(37,43)(38,42)(39,45)(40,46)(41,48);(1,2)(3,7)(4,8)(5,6)(17,19,23,32)(18,24,28,30)(20,35,33,21)(22,36,38,46)(25,26,34,29)(27,40,42,47)(31,37,45,48)(39,43,44,41);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,21,19,23,33,35,32)(18,25,29,24,28,34,26,30)(22,36,37,31,38,46,48,45)(27,40,43,39,42,47,41,44);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,21,23,35)(18,26,28,29)(19,20,32,33)(22,37,38,48)(24,34,30,25)(27,41,42,43)(31,36,45,46)(39,47,44,40);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,22,35,48,23,38,21,37)(18,27,26,41,28,42,29,43)(19,31,20,36,32,45,33,46)(24,39,25,40,30,44,34,47)Polycyclic presentation:
2^9|b^2=i,c^2=fhi,d^2=g,e^2=fghi,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, 6750931);