Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349830 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2651 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)(7,12)(11,16)(14,15)(17,18)(19,29)(20,34)(21,28)(22,27)(23,26)(24,32)(25,37)(30,44)(31,39)(33,40)(35,42)(36,41)(38,45)(43,47)(46,48);(1,3)(2,7)(4,12)(5,14)(6,11)(8,16)(9,15)(10,13)(17,19)(18,24)(20,30)(21,33)(22,32)(23,31)(25,38)(26,40)(27,39)(28,29)(34,48)(35,45)(36,47)(37,46)(41,44)(42,43);(1,2)(3,7)(4,13)(5,8)(6,9)(10,12)(11,15)(14,16)(17,20)(18,25)(19,30)(21,37)(22,36)(23,35)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,47)(33,46)(39,48)(40,44);(3,11)(7,15)(10,16)(12,14)(18,26)(19,31)(25,41)(27,28)(30,45)(32,33)(34,42)(46,47);(1,4,6,5)(2,8,9,13)(3,12,11,14)(7,16,15,10)(20,35)(25,41)(30,45)(34,42)(36,37)(38,44)(43,48)(46,47)Polycyclic presentation:
2^9|e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,d^b=dh,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257076);