Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62125 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,12)(10,13)(11,14)(15,16)(17,18,23,27)(19,28,29,44)(20,25,35,43)(21,26,22,37)(24,31,38,32)(30,45,33,48)(34,41,36,42)(39,47,40,46);(2,6)(7,9)(8,13)(14,15)(17,19,22,32,23,29,21,31)(18,24,26,28,27,38,37,44)(20,33,34,47,35,30,36,46)(25,40,42,48,43,39,41,45);(2,7)(5,12)(6,9)(8,14)(11,16)(13,15)(19,29)(24,38)(28,44)(30,33)(31,32)(39,40)(45,48)(46,47);(1,3)(2,8)(4,10)(5,11)(6,13)(7,14)(9,15)(12,16)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,39)(26,42)(27,43)(28,45)(29,33)(31,47)(32,46)(37,41)(38,40)(44,48);(3,10)(8,13)(11,16)(14,15)(17,21,23,22)(18,26,27,37)(19,31,29,32)(20,34,35,36)(24,28,38,44)(25,41,43,42)(30,46,33,47)(39,48,40,45)Polycyclic presentation:
2^9|a^2=h,b^2=fi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264207);