Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366870 |
| 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 | #61987 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1515 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)(3,12)(4,9)(5,11)(6,10)(7,14)(8,13)(15,16)(17,18,23,28)(19,29,33,38)(20,25,22,27)(21,26,36,44)(24,32,40,30)(31,45,47,42)(34,41,37,43)(35,48,46,39);(1,3)(2,7)(4,15)(5,14)(6,13)(8,9)(10,16)(11,12)(17,19)(18,24)(20,30)(21,35)(22,32)(23,33)(25,38)(26,42)(27,29)(28,40)(31,34)(36,46)(37,47)(39,43)(41,48)(44,45);(1,3)(2,8)(4,13)(5,14)(6,15)(7,9)(10,12)(11,16)(17,20,23,22)(18,25,28,27)(19,30,33,32)(21,34,36,37)(24,38,40,29)(26,41,44,43)(31,46,47,35)(39,42,48,45);(1,4,5,6)(2,9,11,10)(3,13,14,15)(7,16,12,8)(17,21)(18,26)(19,31)(20,34)(22,37)(23,36)(24,39)(25,41)(27,43)(28,44)(29,45)(30,46)(32,35)(33,47)(38,42)(40,48);(1,4,5,6)(2,10,11,9)(3,13,14,15)(7,8,12,16)(17,20,23,22)(18,27,28,25)(19,32,33,30)(21,34,36,37)(24,38,40,29)(26,43,44,41)(31,35,47,46)(39,42,48,45)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=i,e^2=hi,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,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, 6274359);