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 |
| prime radical | #43283 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,14)(4,6,10,16)(5,9,15,13)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,31)(29,32)(33,34,39,42)(35,40,45,37)(36,38,48,41)(43,47,46,44);(1,3)(2,8)(4,13)(5,6)(7,12)(9,10)(11,14)(15,16)(17,19)(18,22)(20,26)(21,27)(23,31)(24,25)(28,32)(29,30)(33,35,39,45)(34,40,42,37)(36,46,48,43)(38,44,41,47);(3,12)(5,15)(8,14)(9,13)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,36,39,48)(34,41,42,38)(35,43,45,46)(37,47,40,44);(1,4)(2,6)(3,13)(5,8)(7,10)(9,12)(11,16)(14,15)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31)(33,37)(34,35)(36,47)(38,46)(39,40)(41,43)(42,45)(44,48);(1,5)(2,9)(3,6)(4,14)(7,15)(8,10)(11,13)(12,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,36)(34,38)(35,43)(37,44)(39,48)(40,47)(41,42)(45,46);(1,6)(2,10)(3,5)(4,11)(7,16)(8,9)(12,15)(13,14)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)(33,38)(34,36)(35,44)(37,43)(39,41)(40,46)(42,48)(45,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8973216);