Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| 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 | #61414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1664 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)(17,18,22,26,23,28,36,27)(19,29,32,42,34,45,33,24)(20,35,21,25,37,44,39,43)(30,47,48,41,31,40,38,46);(3,9)(4,12)(6,13)(7,11)(17,19)(18,24)(20,30)(21,38)(22,33)(23,34)(25,40)(26,45)(27,29)(28,42)(31,37)(32,36)(35,46)(39,48)(41,44)(43,47);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,36,39,23,37,22,21)(18,25,26,44,28,43,27,35)(19,30,32,48,34,31,33,38)(24,40,45,41,42,47,29,46);(3,9)(6,13)(10,15)(14,16)(17,21,22,37,23,39,36,20)(18,25,26,44,28,43,27,35)(19,31,32,38,34,30,33,48)(24,41,29,40,42,46,45,47);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(20,37)(21,39)(25,43)(30,31)(35,44)(38,48)(40,47)(41,46)Polycyclic presentation:
2^9|a^2=fh,c^2=ghi,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104030);