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 | #50051 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,10,6,8)(4,13,14,5)(11,16,15,12)(18,23)(19,26)(25,32)(27,28)(33,34)(35,37)(36,40)(38,42)(39,46)(41,47)(43,48)(44,45);(1,3,7,6)(2,8,9,10)(4,11,14,15)(5,16,13,12)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,31)(30,32)(33,35,38,41)(34,37,42,47)(36,44,46,43)(39,48,40,45);(2,9)(4,14)(8,10)(11,15)(33,36,38,46)(34,39,42,40)(35,43,41,44)(37,45,47,48);(1,4,7,14)(2,5,9,13)(3,11,6,15)(8,16,10,12)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,31,32,29)(33,36)(34,40)(35,44)(37,45)(38,46)(39,42)(41,43)(47,48);(1,5)(2,4)(3,12)(6,16)(7,13)(8,15)(9,14)(10,11)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,35)(34,37)(36,43)(38,41)(39,45)(40,48)(42,47)(44,46);(1,6,7,3)(2,10,9,8)(4,15,14,11)(5,12,13,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,32,28)(33,37)(34,41)(35,42)(36,45)(38,47)(39,44)(40,43)(46,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10066408);