Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351503 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62752 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2620 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,28)(19,29,31,40)(20,34,36,42)(21,27,22,26)(24,32,39,33)(25,37,41,35)(30,38,44,48)(43,46,47,45);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,24)(20,30)(21,32)(22,33)(23,31)(25,38)(26,40)(27,29)(28,39)(34,47)(35,45)(36,44)(37,46)(41,48)(42,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20,23,36)(18,25,28,41)(19,30,31,44)(21,35,22,37)(24,38,39,48)(26,34,27,42)(29,43,40,47)(32,45,33,46);(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,36,37)(24,40,39,29)(25,34,41,42)(30,45,44,46)(38,47,48,43)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=ghi,d^b=dh,f^b=fh,g^b=gi,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258744);