Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357957 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62425 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1779 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)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,18,23,27)(19,28,32,43)(20,25,34,41)(21,26,22,36)(24,30,38,31)(29,42,45,47)(33,39,35,40)(37,46,48,44);(1,2,6,9,7,11,5,10)(3,4,8,12,14,16,13,15)(17,19,21,30,23,32,22,31)(18,24,36,43,27,38,26,28)(20,29,35,46,34,45,33,44)(25,37,39,47,41,48,40,42);(1,3,6,8,7,14,5,13)(2,4,9,12,11,16,10,15);(1,4)(2,8)(3,10)(5,15)(6,12)(7,16)(9,14)(11,13)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,37)(26,40)(27,41)(28,42)(30,46)(31,44)(32,45)(36,39)(38,48)(43,47);(1,5,7,6)(2,9,11,10)(3,13,14,8)(4,12,16,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,43)(25,39,41,40)(29,44,45,46)(37,47,48,42)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=di,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265204);