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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,7,13,16)(10,14,12,15)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32)(33,34,38,39)(35,41,42,36)(37,45,46,40)(43,44,48,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32)(33,35,38,42)(34,36,39,41)(37,43,46,48)(40,44,45,47);(18,24)(20,25)(23,30)(26,31)(34,39)(35,42)(40,45)(43,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32)(33,34)(35,36)(37,40)(38,39)(41,42)(43,44)(45,46)(47,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)(33,37,38,46)(34,40,39,45)(35,43,42,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10073963);