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 | #34589 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4084 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)(17,18,22,23)(19,27,29,24)(20,26,21,25)(28,31,30,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(18,23)(20,21)(24,27)(28,30)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4)(2,5)(3,12)(6,10)(7,9)(8,13)(11,14)(15,16)(17,19)(18,24)(20,30)(21,28)(22,29)(23,27)(25,32)(26,31)(35,41)(36,43)(38,44)(39,46);(1,5,7,6)(2,4,10,9)(3,13,15,14)(8,12,11,16)(17,20,22,21)(18,25,23,26)(19,28,29,30)(24,31,27,32)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48);(1,5,7,6)(2,9,10,4)(3,14,15,13)(8,12,11,16)(17,21,22,20)(18,26,23,25)(19,28,29,30)(24,31,27,32)(35,41)(38,44)(42,47)(45,48);(1,6,7,5)(2,4,10,9)(3,14,15,13)(8,12,11,16)Polycyclic presentation:
2^9|a^2=gi,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, 8011657);