Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351787 |
| 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 | |
2676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,7)(3,11)(4,5)(9,10)(12,13)(14,16)(17,18)(19,29)(20,25)(21,28)(22,27)(23,26)(24,31)(30,44)(32,41)(33,40)(34,42)(35,38)(36,39)(37,47)(43,45)(46,48);(1,2)(3,8)(4,9)(5,7)(6,10)(11,16)(12,15)(13,14)(17,19,23,32)(18,24,26,38)(20,30,33,45)(21,31,22,35)(25,37,40,48)(27,29,28,41)(34,46,36,47)(39,43,42,44);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20)(18,25)(19,30)(21,34)(22,36)(23,33)(24,37)(26,40)(27,39)(28,42)(29,43)(31,46)(32,45)(35,47)(38,48)(41,44);(3,12)(8,15)(11,13)(14,16)(18,26)(20,33)(24,38)(27,28)(29,41)(30,45)(34,36)(46,47);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,16,15,14)(17,21,23,22)(18,27,26,28)(19,31,32,35)(20,34,33,36)(24,29,38,41)(25,39,40,42)(30,46,45,47)(37,43,48,44)Polycyclic presentation:
2^9|b^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258993);