Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366710 |
| 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 |
| prime radical | #61937 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1504 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)(3,13)(4,14)(5,11)(6,10)(7,12)(8,15)(9,16)(18,24)(21,22)(25,38)(26,39)(27,28)(29,42)(31,36)(33,41)(34,37)(40,44)(43,48)(45,46);(1,3,7,8)(2,4,12,16)(5,14,6,9)(10,13,11,15)(17,18,23,28)(19,29,32,25)(20,33,35,39)(21,24,22,27)(26,37,41,34)(30,44,47,48)(31,38,36,42)(40,46,43,45);(1,4,7,16)(2,8,12,3)(5,15,6,13)(9,10,14,11)(17,19)(18,25)(20,30)(21,36)(22,31)(23,32)(24,38)(26,40)(27,42)(28,29)(33,48)(34,46)(35,47)(37,45)(39,44)(41,43);(1,4,7,16)(2,9,12,14)(3,11,8,10)(5,13,6,15)(17,20,23,35)(18,26,28,41)(19,30,32,47)(21,34,22,37)(24,39,27,33)(25,40,29,43)(31,45,36,46)(38,44,42,48);(1,5,7,6)(2,10,12,11)(3,14,8,9)(4,13,16,15)(17,21,23,22)(18,27,28,24)(19,31,32,36)(20,34,35,37)(25,38,29,42)(26,33,41,39)(30,45,47,46)(40,44,43,48)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=h,e^2=h,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=ch,d^b=df,e^b=ehi,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, 6273972);