Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1563 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)(11,15)(12,16)(13,14)(17,18,22,27,23,28,20,25)(19,29,30,38,32,42,36,24)(21,26,34,40,35,41,33,39)(31,43,44,48,46,47,45,37);(9,10)(11,16)(12,15)(13,14)(17,19,23,32)(18,24,28,38)(20,30,22,36)(21,31,35,46)(25,29,27,42)(26,37,41,48)(33,44,34,45)(39,43,40,47);(2,5)(6,8)(10,14)(15,16)(17,20,23,22)(18,25,28,27)(19,30,32,36)(21,34,35,33)(24,29,38,42)(26,40,41,39)(31,45,46,44)(37,47,48,43);(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,31)(20,33)(22,34)(23,35)(24,37)(25,39)(27,40)(28,41)(29,43)(30,44)(32,46)(36,45)(38,48)(42,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(21,35)(26,41)(31,46)(33,34)(37,48)(39,40)(43,47)(44,45)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=g,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, 6392500);