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 | |
4216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,12,6)(4,10,14,15)(5,11,13,16)(17,18)(19,27)(20,30)(21,26)(22,24)(23,25)(28,32)(29,31)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(1,3)(2,8)(4,13)(5,14)(6,9)(7,12)(10,16)(11,15)(17,19,23,22)(18,24,25,27)(20,28,26,31)(21,32,30,29)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,9)(3,12)(4,14)(11,16)(18,25)(20,26)(24,27)(28,31)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,4,7,14)(2,10,9,15)(3,13,12,5)(6,11,8,16)(17,20)(18,21)(19,28)(22,31)(23,26)(24,32)(25,30)(27,29)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,5,7,13)(2,11,9,16)(3,4,12,14)(6,15,8,10)(17,21)(18,26)(19,29)(20,25)(22,32)(23,30)(24,28)(27,31)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,6)(2,12)(3,9)(4,11)(5,10)(7,8)(13,15)(14,16)(17,22,23,19)(18,27,25,24)(20,31,26,28)(21,29,30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=ghi,d^2=gh,e^2=gh,f^2=hi,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, 10103071);