Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1425 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18,21,26,22,24,34,38)(19,27,29,35,30,40,33,23)(20,25,32,39,31,37,44,47)(28,41,43,46,42,48,45,36);(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,24,35)(20,28,31,42)(21,33,34,29)(25,36,37,46)(26,40,38,27)(32,45,44,43)(39,48,47,41);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(18,24)(20,31)(23,35)(26,38)(27,40)(28,42)(32,44)(43,45);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,22,31)(18,25,24,37)(19,28,30,42)(21,32,34,44)(23,36,35,46)(26,39,38,47)(27,41,40,48)(29,43,33,45);(9,11,12,15)(10,14,13,16)(20,31)(25,37)(28,42)(32,44)(36,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403020);