Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| 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 | |
1581 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)(19,24)(20,35)(21,26)(22,28)(23,29)(25,31)(27,40)(30,45)(32,48)(33,41)(34,42)(36,44)(37,47)(38,43)(39,46);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,19)(18,24)(20,21)(22,32)(23,34)(25,43)(26,35)(27,41)(28,48)(29,42)(30,36)(31,38)(33,40)(37,39)(44,45)(46,47);(17,20,22,36,23,37,40,31)(18,25,28,35,29,44,27,47)(19,30,33,21,34,38,32,39)(24,26,41,43,42,46,48,45);(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)(17,21,22,38,23,39,40,30)(18,26,28,43,29,46,27,45)(19,31,33,20,34,36,32,37)(24,25,41,35,42,44,48,47)Polycyclic presentation:
2^9|a^2=f,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,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719059);