Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | #3062 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,23,29,24,30,22,28)(19,31,32,25,33,44,41,45)(20,34,36,43,38,46,37,26)(21,27,40,48,35,42,39,47);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,22,41,24,33,23,32)(18,25,29,44,30,45,28,31)(20,35,36,39,38,21,37,40)(26,27,46,47,43,42,34,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,21)(22,36)(23,37)(24,38)(25,42)(27,45)(28,34)(29,46)(30,43)(31,47)(32,39)(33,35)(40,41)(44,48);(3,10)(6,9)(11,16)(14,15)(17,21,24,35)(18,27,30,42)(19,20,33,38)(22,39,23,40)(25,43,45,26)(28,47,29,48)(31,34,44,46)(32,36,41,37);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,24,23)(18,28,30,29)(19,32,33,41)(20,36,38,37)(21,39,35,40)(25,44,45,31)(26,34,43,46)(27,47,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=fh,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090007);