Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6361928 |
| 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 | |
1500 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)(3,10)(4,9)(5,13)(6,8)(7,12)(11,16)(14,15)(17,18,24,29)(19,30,34,46)(20,21,26,39)(22,28,23,27)(25,41,45,33)(31,32,35,48)(36,42,40,38)(37,47,43,44);(3,11)(7,15)(10,16)(12,14)(17,19,24,34)(18,25,29,45)(20,35,26,31)(21,37,39,43)(22,33,23,41)(27,30,28,46)(32,42,48,38)(36,47,40,44);(1,3)(2,7)(4,12)(5,14)(6,11)(8,15)(9,16)(10,13)(17,20,24,26)(18,21,29,39)(19,31,34,35)(22,40,23,36)(25,43,45,37)(27,42,28,38)(30,32,46,48)(33,47,41,44);(2,8)(7,15)(9,13)(10,16)(17,21)(18,26)(19,32)(20,29)(22,38)(23,42)(24,39)(25,44)(27,40)(28,36)(30,35)(31,46)(33,43)(34,48)(37,41)(45,47);(1,4,6,5)(2,9,8,13)(3,12,11,14)(7,16,15,10)(17,22,24,23)(18,27,29,28)(19,33,34,41)(20,36,26,40)(21,38,39,42)(25,30,45,46)(31,44,35,47)(32,43,48,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269205);