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 | #3150 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1632 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)(17,18,23,40,24,29,22,28)(19,30,39,25,34,43,33,44)(20,35,37,26,36,46,21,27)(31,41,38,48,32,45,47,42);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,31)(21,38)(22,39)(23,33)(24,34)(26,41)(27,45)(28,43)(29,44)(30,40)(32,36)(35,48)(37,47)(42,46);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,23,37,24,36,22,21)(18,26,28,35,29,27,40,46)(19,31,33,47,34,32,39,38)(25,41,43,48,44,45,30,42);(3,9)(6,13)(10,15)(14,16)(17,21,22,36,24,37,23,20)(18,27,28,46,29,26,40,35)(19,32,33,38,34,31,39,47)(25,42,30,45,44,48,43,41);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(20,36)(21,37)(26,27)(31,32)(35,46)(38,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=fhi,c^2=gh,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, 7104029);