Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| 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 | #30621 | 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) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,25)(20,21)(22,26)(23,27)(24,28)(29,33)(30,39)(31,44)(32,40)(34,35)(36,41)(37,42)(38,43)(45,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,33)(21,29)(22,36)(23,31)(24,32)(26,41)(27,44)(28,40)(30,37)(34,47)(35,46)(38,45)(39,42)(43,48);(17,20,23,18,24,27,21,28)(19,25,29,33,32,40,31,44)(22,34,37,26,38,42,35,43)(30,39,36,41,46,47,45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,24,23)(18,20,28,27)(19,29,32,31)(22,35,38,37)(25,33,40,44)(26,34,43,42)(30,36,46,45)(39,41,47,48);(3,11)(6,9)(10,16)(14,15)(17,22,23,37,24,38,21,35)(18,26,27,42,28,43,20,34)(19,30,29,36,32,46,31,45)(25,39,33,41,40,47,44,48)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=fi,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715963);