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 | #37701 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(17,18,22,27)(19,28,31,39)(20,33,35,41)(21,25,29,26)(23,36,38,32)(24,40,42,34)(30,44,45,37)(43,47,48,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,29,36,22,31,21,32)(18,23,25,39,27,38,26,28)(20,30,40,47,35,45,34,46)(24,37,41,48,42,44,33,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,29,40,22,35,21,34)(18,24,25,41,27,42,26,33)(19,30,36,47,31,45,32,46)(23,37,39,48,38,44,28,43);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,31)(23,38)(28,39)(30,45)(32,36)(37,44)(43,48)(46,47);(11,15)(14,16)(17,21,22,29)(18,25,27,26)(19,32,31,36)(20,34,35,40)(23,39,38,28)(24,41,42,33)(30,46,45,47)(37,48,44,43)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fi,d^2=fgi,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, 6464642);