Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352941 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62963 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,7)(5,9)(6,14)(8,12)(11,15)(13,16)(17,18,23,27)(19,28,31,37)(20,25,22,35)(21,26,33,41)(24,36,39,29)(30,42,44,47)(32,40,34,45)(38,46,48,43);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,39)(28,35)(32,43)(33,44)(34,46)(40,47)(41,48)(42,45);(1,4,7,2)(3,11,13,8)(5,14,6,9)(10,15,16,12)(17,20,23,22)(18,25,27,35)(19,29,31,36)(21,32,33,34)(24,37,39,28)(26,40,41,45)(30,43,44,46)(38,47,48,42);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(17,21,23,33)(18,26,27,41)(19,30,31,44)(20,32,22,34)(24,38,39,48)(25,40,35,45)(28,42,37,47)(29,43,36,46);(21,33)(26,41)(30,44)(32,34)(38,48)(40,45)(42,47)(43,46)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260238);