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 | |
4085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,10)(5,12,13,8)(11,16,15,14)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(2,4)(3,9)(8,12)(11,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(2,4)(3,9)(8,12)(11,15)(18,22)(23,25)(24,28)(30,31)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9928028);