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 | |
4239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,14,8)(4,9,10,5)(12,16,15,13)(18,22)(23,25)(24,29)(30,31)(33,34)(35,39)(36,45)(37,40)(38,42)(41,44)(43,48)(46,47);(2,7)(4,10)(8,11)(12,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35)(34,39)(36,46)(37,43)(38,44)(40,48)(41,42)(45,47);(2,7)(5,9)(8,11)(13,16)(20,28)(24,29)(26,32)(30,31)(33,36,38,37)(34,40,42,45)(35,43,44,46)(39,47,41,48);(1,3,6,14)(2,8,7,11)(4,15,10,12)(5,13,9,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,37,38,36)(34,40,42,45)(35,43,44,46)(39,48,41,47);(1,4,6,10)(2,9,7,5)(3,12,14,15)(8,13,11,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,34)(35,41)(36,40)(37,45)(38,42)(39,44)(43,48)(46,47);(1,5,6,9)(2,10,7,4)(3,13,14,16)(8,12,11,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,35)(34,41)(36,46)(37,43)(38,44)(39,42)(40,47)(45,48)Polycyclic presentation:
2^9|a^2=hi,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8805713);