Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806827 |
| 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 | C22×C4 | C25 |
| prime radical | #60694 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,9)(3,10,12,7)(4,13,5,8)(11,15,14,16)(17,18,24,29)(19,25,33,44)(20,34,36,46)(21,37,39,27)(22,40,23,28)(26,31,47,35)(30,45,38,48)(32,43,41,42);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,19)(18,25)(20,21)(22,32)(23,41)(24,33)(26,45)(27,46)(28,42)(29,44)(30,35)(31,38)(34,37)(36,39)(40,43)(47,48);(17,20,23,35,24,36,22,31)(18,26,28,46,29,47,40,34)(19,30,32,21,33,38,41,39)(25,27,43,45,44,37,42,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21,23,38,24,39,22,30)(18,27,28,45,29,37,40,48)(19,31,32,20,33,35,41,36)(25,26,43,46,44,47,42,34)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=h,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719069);