Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357939 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60801 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(4,11)(6,12)(8,14)(10,15)(13,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,42)(30,43)(31,38)(33,39)(34,40)(35,41)(37,44)(45,47)(46,48);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,38,36,43)(20,29,34,44,33,45,35,46)(25,37,41,42,39,48,40,47);(1,3,4,9,5,10,11,15)(2,6,8,12,7,13,14,16)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,37)(26,41)(27,39)(28,42)(30,45)(31,46)(32,44)(36,40)(38,48)(43,47);(3,10)(6,13)(9,15)(12,16)(19,30)(20,33)(24,38)(25,39)(28,43)(31,32)(34,35)(40,41);(2,7)(6,13)(8,14)(12,16)(17,21,23,22)(18,26,27,36)(19,31,30,32)(20,34,33,35)(24,28,38,43)(25,40,39,41)(29,44,45,46)(37,47,48,42)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,e^b=ei,d^c=dhi,e^c=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265140);