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 | |
4180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,10,7)(5,12,13,8)(11,14,16,15)(21,29)(25,30)(26,31)(27,32)(33,34,39,44)(35,42,37,40)(36,43,38,41)(45,47,46,48);(2,4)(7,9)(8,12)(14,15)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(33,35)(34,40)(36,45)(37,39)(38,46)(41,47)(42,44)(43,48);(3,10)(7,9)(11,16)(14,15)(18,23)(19,24)(25,30)(26,31)(33,36,39,38)(34,41,44,43)(35,45,37,46)(40,47,42,48);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,37,39,35)(34,42,44,40)(36,45,38,46)(41,47,43,48);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,38,39,36)(34,43,44,41)(35,45,37,46)(40,47,42,48);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019738);