Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356733 |
| 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 | #61982 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(4,12)(7,13)(8,14)(9,16)(11,15)(17,18,23,28)(19,29,31,40)(20,34,36,41)(21,26,22,27)(24,33,39,32)(25,37,42,35)(30,44,45,38)(43,47,48,46);(1,2,4,6,5,8,12,14)(3,9,11,16,10,7,15,13)(17,19,21,32,23,31,22,33)(18,24,27,29,28,39,26,40)(20,30,35,46,36,45,37,47)(25,38,34,43,42,44,41,48);(17,20,23,36)(18,25,28,42)(19,30,31,45)(21,35,22,37)(24,38,39,44)(26,41,27,34)(29,43,40,48)(32,46,33,47);(1,3)(2,7)(4,11)(5,10)(6,13)(8,9)(12,15)(14,16)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(3,10)(7,9)(11,15)(13,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,36,37)(24,40,39,29)(25,41,42,34)(30,46,45,47)(38,48,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,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, 6263987);