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 | |
1572 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)(3,6)(4,7)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18,22,25)(19,27,30,43)(20,24,33,41)(21,34,37,26)(23,35,39,29)(28,45,46,40)(31,48,47,38)(32,44,36,42);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,22,30)(18,23,25,39)(20,31,33,47)(21,35,37,29)(24,40,41,45)(26,43,34,27)(28,36,46,32)(38,44,48,42);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,29)(28,38)(30,39)(31,40)(32,42)(33,41)(34,37)(35,43)(36,44)(45,47)(46,48);(17,20,22,33)(18,24,25,41)(19,28,30,46)(21,36,37,32)(23,38,39,48)(26,44,34,42)(27,45,43,40)(29,47,35,31);(2,8)(3,10)(7,9)(11,16)(18,25)(23,39)(24,41)(26,34)(27,43)(38,48)(40,45)(42,44)Polycyclic presentation:
2^9|a^2=h,b^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, 6274741);