Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356633 |
| 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 | #3482 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,4)(3,10,13,16)(5,9,6,14)(8,15,11,12)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,42)(30,43)(31,37)(33,39)(34,41)(35,40)(38,45)(44,48)(46,47);(17,19,21,31,23,29,22,32)(18,24,36,42,27,37,26,28)(20,30,35,46,34,44,33,45)(25,38,39,48,41,47,40,43);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(19,29)(24,37)(28,42)(30,44)(31,32)(38,47)(43,48)(45,46);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,20,23,34)(18,25,27,41)(19,30,29,44)(21,35,22,33)(24,38,37,47)(26,40,36,39)(28,43,42,48)(31,46,32,45);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(17,21,23,22)(18,26,27,36)(19,31,29,32)(20,33,34,35)(24,28,37,42)(25,39,41,40)(30,45,44,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=i,b^2=fh,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,g^c=gi,e^d=ehi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263873);