Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(4,12)(7,10)(8,13)(17,18,23,27)(19,28,25,21)(20,24,30,32)(22,26,29,31)(33,34,38,43)(35,40,44,39)(36,45,47,41)(37,46,48,42);(1,2)(3,10)(4,13)(5,6)(7,9)(8,12)(11,15)(14,16)(17,19,23,25)(18,21,27,28)(20,22,30,29)(24,31,32,26)(33,34)(35,39)(36,41)(37,42)(38,43)(40,44)(45,47)(46,48);(2,6)(3,9)(8,13)(11,14)(20,30)(22,29)(24,32)(26,31)(33,35,38,44)(34,39,43,40)(36,42,47,46)(37,45,48,41);(1,3)(2,7)(4,14)(5,9)(6,10)(8,16)(11,12)(13,15)(17,20,23,30)(18,24,27,32)(19,29,25,22)(21,26,28,31)(33,35)(34,40)(36,46)(37,41)(38,44)(39,43)(42,47)(45,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,15)(9,14)(10,16)(17,21,23,28)(18,25,27,19)(20,31,30,26)(22,24,29,32)(33,36,38,47)(34,41,43,45)(35,42,44,46)(37,40,48,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,22,23,29)(18,26,27,31)(19,30,25,20)(21,24,28,32)(33,37,38,48)(34,42,43,46)(35,41,44,45)(36,40,47,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840398);