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 | #38824 | C2 |
| 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:
(1,2)(3,8)(4,6)(5,10)(7,9)(11,13)(12,15)(14,16)(18,23)(19,26)(25,32)(27,28)(33,34,38,42)(35,43,45,39)(36,44,46,40)(37,41,48,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,15,16,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,31)(30,32)(33,35,38,45)(34,39,42,43)(36,41,46,47)(37,40,48,44);(2,9)(3,4)(10,16)(11,12)(33,36,38,46)(34,40,42,44)(35,41,45,47)(37,39,48,43);(1,4,7,3)(2,6,9,8)(5,11,14,12)(10,13,16,15)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,31,32,29)(33,36)(34,40)(35,41)(37,43)(38,46)(39,48)(42,44)(45,47);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,15,8,13)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,37)(34,41)(35,44)(36,39)(38,48)(40,45)(42,47)(43,46);(1,6)(2,3)(4,9)(5,13)(7,8)(10,12)(11,16)(14,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,32,28)(33,34)(35,39)(36,40)(37,47)(38,42)(41,48)(43,45)(44,46)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=ghi,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, 10103158);