Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #61898 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1541 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)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(17,18,23,27)(19,28,31,39)(20,25,22,36)(21,33,35,26)(24,37,41,29)(30,43,47,45)(32,44,38,42)(34,40,46,48);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,23,31)(18,24,27,41)(20,29,22,37)(21,34,35,46)(25,39,36,28)(26,43,33,45)(30,38,47,32)(40,42,48,44);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,20,23,22)(18,25,27,36)(19,29,31,37)(21,32,35,38)(24,39,41,28)(26,42,33,44)(30,46,47,34)(40,43,48,45);(17,21,23,35)(18,26,27,33)(19,30,31,47)(20,32,22,38)(24,40,41,48)(25,42,36,44)(28,45,39,43)(29,46,37,34);(2,9)(7,13)(8,16)(11,15)(18,27)(24,41)(25,36)(26,33)(28,39)(40,48)(42,44)(43,45)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ehi,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269781);