Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368046 |
| 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 | #62464 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1496 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)(3,4)(5,13)(6,9)(7,12)(8,11)(10,16)(14,15)(17,18,22,25)(19,23,28,38)(20,30,32,40)(21,33,35,26)(24,34,39,31)(27,41,42,48)(29,45,44,37)(36,46,47,43);(1,3)(2,7)(4,13)(5,15)(6,11)(8,10)(9,14)(12,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,38)(26,37)(30,41)(31,43)(32,42)(33,45)(34,46)(35,44)(39,47)(40,48);(1,3)(2,8)(4,9)(5,12)(6,11)(7,10)(13,14)(15,16)(17,20,22,32)(18,24,25,39)(19,27,28,42)(21,34,35,31)(23,36,38,47)(26,40,33,30)(29,46,44,43)(37,48,45,41);(1,4,6,8)(2,3,9,11)(5,7,16,14)(10,15,13,12);(2,9)(3,11)(10,13)(12,15)(18,25)(19,28)(24,39)(26,33)(27,42)(29,44)(30,40)(43,46)Polycyclic presentation:
2^9|a^2=i,c^2=i,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=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275410);