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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4001 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,31)(28,30)(29,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,29)(18,23,26,32)(20,27,21,28)(24,30,25,31)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20,22,21)(18,24,26,25)(19,27,29,28)(23,30,32,31)(35,41)(36,43)(38,44)(39,46);(2,7)(6,9)(8,14)(13,15)(17,20,22,21)(18,25,26,24)(19,28,29,27)(23,30,32,31)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48);(3,10)(6,9)(11,16)(13,15)(17,21,22,20)(18,25,26,24)(19,28,29,27)(23,31,32,30)(35,41)(38,44)(42,47)(45,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=g,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, 8011615);