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 | |
3990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(18,20)(19,22)(21,29)(31,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,18,23,20)(19,24,22,27)(21,25,29,28)(26,31,30,32)(36,44)(39,46)(42,48)(45,47);(2,6)(7,10)(8,12)(14,16)(17,19,23,22)(18,24,20,27)(21,26,29,30)(25,31,28,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,24,22,27)(21,28,29,25)(26,31,30,32);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,19)(18,24,20,27)(21,30,29,26)(25,31,28,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662793);