Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #53672 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,11,15,8)(4,5,9,6)(12,14,16,13)(17,18,23,26)(19,22,25,21)(20,28,30,24)(27,31,32,29)(33,34,38,37)(35,39,43,42)(36,40,45,46)(41,47,44,48);(1,3,7,15)(2,8,10,11)(4,12,9,16)(5,13,6,14)(17,19,23,25)(18,21,26,22)(20,27,30,32)(24,31,28,29)(33,35,38,43)(34,39,37,42)(36,44,45,41)(40,48,46,47);(1,4)(2,5)(3,12)(6,10)(7,9)(8,13)(11,14)(15,16)(17,20)(18,24)(19,27)(21,31)(22,29)(23,30)(25,32)(26,28)(33,36,38,45)(34,40,37,46)(35,41,43,44)(39,47,42,48);(1,5,7,6)(2,4,10,9)(3,13,15,14)(8,12,11,16)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,37,38,34)(35,42,43,39)(36,40,45,46)(41,47,44,48);(1,5,7,6)(2,9,10,4)(3,14,15,13)(8,12,11,16)(17,22,23,21)(18,19,26,25)(20,29,30,31)(24,27,28,32)(35,43)(39,42)(41,44)(47,48);(1,6,7,5)(2,4,10,9)(3,14,15,13)(8,12,11,16)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=i,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012360);