Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367534 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,15)(4,7,12,9)(11,13,16,14)(17,18)(19,27)(20,23)(21,25)(22,33)(24,30)(26,39)(28,37)(29,45)(31,43)(32,36)(34,46)(35,44)(38,42)(40,48)(41,47);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,23,31)(18,24,20,28)(21,34,36,48)(22,37,39,30)(25,41,32,45)(26,43,33,27)(29,38,47,35)(40,44,46,42);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20,23,18)(19,28,31,24)(21,32,36,25)(22,33,39,26)(27,30,43,37)(29,46,47,40)(34,45,48,41)(35,44,38,42);(17,21,23,36)(18,25,20,32)(19,29,31,47)(22,38,39,35)(24,40,28,46)(26,44,33,42)(27,45,43,41)(30,48,37,34);(2,8)(3,10)(7,9)(11,16)(18,20)(24,28)(25,32)(26,33)(27,43)(40,46)(41,45)(42,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274776);