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 | |
3979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(5,13)(7,10)(11,15)(18,20)(19,21)(22,30)(31,32)(33,34)(35,43)(36,47)(37,48)(38,42)(39,40)(41,44)(45,46);(2,4)(3,9)(8,12)(11,15)(17,18)(19,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(34,40)(35,44)(36,48)(42,46);(1,2,6,4)(3,10,9,7)(5,8,13,12)(11,16,15,14)(17,19,23,21)(18,24,20,27)(22,26,30,29)(25,31,28,32)(33,35,39,44)(34,41,40,43)(36,45,48,38)(37,46,47,42);(1,3,6,9)(2,7,4,10)(5,11,13,15)(8,14,12,16)(17,20,23,18)(19,24,21,27)(22,28,30,25)(26,31,29,32)(33,36)(34,37)(35,45)(38,44)(39,48)(40,47)(41,46)(42,43);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,21,23,19)(18,24,20,27)(22,29,30,26)(25,31,28,32)(33,37)(34,36)(35,46)(38,43)(39,47)(40,48)(41,45)(42,44);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,22,23,30)(18,25,20,28)(19,26,21,29)(24,31,27,32)(33,38,39,45)(34,42,40,46)(35,36,44,48)(37,43,47,41)Polycyclic presentation:
2^9|c^2=h,d^2=gi,e^2=gi,f^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, 8021576);