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 | #2006 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1574 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)(17,18,23,27,24,29,22,28)(19,30,25,20,33,35,32,36)(21,26,39,43,40,44,38,45)(31,46,41,34,47,42,37,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,23,25,24,33,22,32)(18,20,28,30,29,36,27,35)(21,37,38,47,40,41,39,31)(26,42,43,48,44,46,45,34);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,24,36)(18,25,29,32)(19,28,33,27)(21,34,40,48)(22,35,23,30)(26,41,44,37)(31,45,47,43)(38,42,39,46);(3,10)(6,9)(11,16)(14,15)(17,21,24,40)(18,26,29,44)(19,31,33,47)(20,34,36,48)(22,38,23,39)(25,41,32,37)(27,43,28,45)(30,46,35,42);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,24,23)(18,27,29,28)(19,32,33,25)(20,35,36,30)(21,38,40,39)(26,43,44,45)(31,37,47,41)(34,42,48,46)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=i,d^2=i,e^2=hi,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, 7090012);