Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 | #61199 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,23,27)(19,29,32,44)(20,35,37,43)(21,26,30,42)(22,41,36,28)(24,34,45,33)(25,31,46,38)(39,48,40,47);(3,9)(4,12)(6,13)(7,11)(17,19)(18,24)(20,21)(22,34)(23,32)(25,26)(27,45)(28,44)(29,41)(30,37)(31,39)(33,36)(35,48)(38,40)(42,46)(43,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,36,31,23,37,22,38)(18,25,41,43,27,46,28,35)(19,30,33,40,32,21,34,39)(24,42,29,48,45,26,44,47);(3,9)(6,13)(10,15)(14,16)(17,21,23,30)(18,26,27,42)(19,31,32,38)(20,33,37,34)(22,40,36,39)(24,43,45,35)(25,29,46,44)(28,48,41,47);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(19,32)(20,37)(26,42)(28,41)(31,38)(33,34)(47,48)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7081038);