Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365522 |
| 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 | #61896 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,7)(4,5)(8,13)(9,15)(10,16)(11,14)(17,18,23,28)(19,24,31,39)(20,33,35,42)(21,27,22,26)(25,36,41,34)(29,43,45,48)(30,40,32,38)(37,46,47,44);(1,2)(3,10)(4,7)(5,13)(6,8)(9,11)(12,15)(14,16)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,37)(26,38)(27,40)(28,39)(33,43)(34,44)(35,45)(36,46)(41,47)(42,48);(2,8)(7,13)(9,16)(10,15)(17,20,23,35)(18,25,28,41)(19,29,31,45)(21,34,22,36)(24,37,39,47)(26,33,27,42)(30,44,32,46)(38,43,40,48);(1,3)(2,9)(4,11)(5,14)(6,12)(7,15)(8,16)(10,13)(17,21,23,22)(18,26,28,27)(19,30,31,32)(20,34,35,36)(24,38,39,40)(25,33,41,42)(29,44,45,46)(37,43,47,48);(1,4,6,5)(2,7,8,13)(3,11,12,14)(9,15,16,10)(19,31)(24,39)(29,45)(30,32)(37,47)(38,40)(43,48)(44,46)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,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=ei,f^b=fh,g^c=gi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272846);