Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C42 | C25 |
| commutator, socle | C22 | C23×C42 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,33)(28,35)(29,34)(30,36)(31,37)(32,38)(39,43)(40,44)(41,45)(42,46)(47,48);(2,6)(7,12)(8,13)(14,16)(17,19,21,28,22,29,32,41)(18,23,26,35,25,34,38,45)(20,27,31,40,30,39,42,47)(24,33,37,44,36,43,46,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,36)(26,37)(28,40)(29,39)(32,42)(34,43)(35,44)(38,46)(41,47)(45,48);(2,6)(3,9)(8,13)(10,15)(20,30)(24,36)(27,39)(31,42)(33,43)(37,46)(40,47)(44,48);(3,9)(7,12)(10,15)(14,16)(18,25)(23,34)(24,36)(26,38)(33,43)(35,45)(37,46)(44,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,g^2=i,b^a=bh,d^a=dh,e^a=ei,d^c=dhi,e^c=eh152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249702);