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 | #60926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1592 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)(17,18,21,28,22,26,37,27)(19,29,32,23,33,40,46,39)(20,25,35,41,36,43,34,24)(30,45,31,44,48,38,47,42);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,21,32,22,33,37,46)(18,23,27,29,26,39,28,40)(20,30,34,47,36,48,35,31)(24,38,25,42,41,45,43,44);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,24)(19,30)(21,34)(22,36)(23,38)(25,27)(26,41)(28,43)(29,42)(31,46)(32,47)(33,48)(35,37)(39,45)(40,44);(3,10)(6,9)(11,16)(14,15)(17,20)(18,25)(19,31)(21,35)(22,36)(23,38)(24,27)(26,43)(28,41)(29,44)(30,46)(32,48)(33,47)(34,37)(39,45)(40,42);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,26)(23,39)(24,41)(25,43)(27,28)(29,40)(38,45)(42,44)Polycyclic presentation:
2^9|a^2=g,b^2=fh,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, 7090005);