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 | #37700 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1629 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)(19,28,30,39)(20,32,34,41)(21,25,35,26)(23,36,38,31)(24,40,42,33)(29,44,45,37)(43,47,48,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,31,22,30,35,36)(18,23,26,28,27,38,25,39)(20,29,33,46,34,45,40,47)(24,37,32,43,42,44,41,48);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,21,33,22,34,35,40)(18,24,26,32,27,42,25,41)(19,29,31,46,30,45,36,47)(23,37,28,43,38,44,39,48);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,30)(23,38)(28,39)(29,45)(31,36)(37,44)(43,48)(46,47);(11,15)(14,16)(17,21,22,35)(18,25,27,26)(19,31,30,36)(20,33,34,40)(23,39,38,28)(24,41,42,32)(29,46,45,47)(37,48,44,43)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=f,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, 6464630);