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 | |
4088 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)(3,8)(4,7)(5,9)(6,12)(10,15)(11,13)(14,16)(17,18,23,22)(19,27,30,24)(20,25,29,31)(21,26,28,32)(33,34)(35,42)(36,46)(37,40)(38,41)(39,45)(43,44)(47,48);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,26)(31,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(2,4)(3,9)(10,12)(11,16)(17,20,23,29)(18,25,22,31)(19,28,30,21)(24,26,27,32)(33,36,38,40)(34,37,41,46)(35,43,45,47)(39,44,42,48);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,26)(27,32)(33,37,38,46)(34,40,41,36)(35,44,45,48)(39,47,42,43);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,25,27,31)(33,34)(35,42)(36,37)(38,41)(39,45)(40,46)(43,48)(44,47);(1,6,7,15)(2,10,4,12)(3,11,9,16)(5,13,8,14)(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,26,28,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113765);