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 | #61997 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,4)(3,10,11,16)(5,9,6,15)(8,12,14,13)(17,18,23,28)(19,29,33,40)(20,34,37,43)(21,26,22,27)(24,31,41,32)(25,38,44,36)(30,42,35,39)(45,46,48,47);(1,3,6,13,7,11,5,12)(2,8,9,10,4,14,15,16)(17,19,22,31,23,33,21,32)(18,24,26,40,28,41,27,29)(20,35,38,47,37,30,36,46)(25,42,43,45,44,39,34,48);(3,11)(8,14)(10,16)(12,13)(17,20,23,37)(18,25,28,44)(19,30,33,35)(21,36,22,38)(24,39,41,42)(26,43,27,34)(29,45,40,48)(31,46,32,47);(1,4,7,2)(3,8,11,14)(5,9,6,15)(10,12,16,13)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,29,41,40)(25,43,44,34)(30,46,35,47)(39,45,42,48);(1,5,7,6)(2,9,4,15)(3,12,11,13)(8,10,14,16)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,d^2=h,e^2=hi,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, 6264221);