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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3833 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)(3,8,11,15)(4,6,13,16)(5,10,12,14)(18,23)(19,26)(25,32)(27,28)(33,34,38,39)(35,36,41,40)(37,45,47,42)(43,46,48,44);(1,3,7,11)(2,8,9,15)(4,12,13,5)(6,14,16,10)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,31)(30,32)(34,39)(36,40)(42,45)(44,46);(2,9)(3,11)(4,13)(10,14)(33,35,38,41)(34,40,39,36)(37,43,47,48)(42,46,45,44);(1,4)(2,6)(3,12)(5,11)(7,13)(8,14)(9,16)(10,15)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,31,32,29)(33,35)(34,40)(36,39)(37,43)(38,41)(42,46)(44,45)(47,48);(1,5,7,12)(2,10,9,14)(3,13,11,4)(6,8,16,15)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,36,38,40)(34,41,39,35)(37,46,47,44)(42,43,45,48);(1,6)(2,4)(3,14)(5,15)(7,16)(8,12)(9,13)(10,11)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,32,28)(33,37,38,47)(34,42,39,45)(35,43,41,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=ghi,e^2=gi,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, 10088237);