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 | #23591 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,22,27,23,28,21,38)(19,29,34,41,32,47,33,24)(20,26,36,44,37,43,39,25)(30,40,31,46,35,42,48,45);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,22,34,23,32,21,33)(18,24,38,47,28,41,27,29)(20,35,39,31,37,30,36,48)(25,42,26,45,44,40,43,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,23,37)(18,25,28,44)(19,30,32,35)(21,36,22,39)(24,40,41,42)(26,27,43,38)(29,45,47,46)(31,34,48,33);(3,10)(6,9)(11,16)(14,15)(17,20)(18,26)(19,31)(21,39)(22,36)(23,37)(24,40)(25,38)(27,44)(28,43)(29,46)(30,33)(32,48)(34,35)(41,42)(45,47);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(19,32)(24,41)(29,47)(30,35)(31,48)(33,34)(40,42)(45,46)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^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, 7090010);