Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1496 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)(17,18,23,28)(19,29,31,40)(20,34,37,41)(21,26,22,27)(24,33,39,32)(25,35,42,36)(30,38,44,48)(43,45,47,46);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,33,23,31,21,32)(18,24,26,40,28,39,27,29)(20,30,36,46,37,44,35,45)(25,38,34,47,42,48,41,43);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,21,35,23,37,22,36)(18,25,27,41,28,42,26,34)(19,30,32,45,31,44,33,46)(24,38,29,43,39,48,40,47);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46);(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,40,39,29)(25,34,42,41)(30,45,44,46)(38,47,48,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6447670);