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 | #50835 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4042 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)(17,18,23,28)(19,22,27,24)(20,25,30,32)(21,26,29,31)(33,34)(35,44)(36,46)(37,42)(38,43)(39,40)(41,47)(45,48);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,26)(31,32)(33,35)(34,37)(36,45)(38,46)(39,44)(40,42)(41,48)(43,47);(2,4)(3,9)(10,12)(11,16)(17,20,23,30)(18,25,28,32)(19,29,27,21)(22,31,24,26)(33,36,40,47)(34,41,39,46)(35,45,42,43)(37,48,44,38);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,26)(28,32)(33,37,40,44)(34,42,39,35)(36,38,47,48)(41,45,46,43);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,21,23,29)(18,26,28,31)(19,30,27,20)(22,32,24,25)(33,38)(34,43)(35,41)(36,37)(39,45)(40,48)(42,46)(44,47);(1,6,7,15)(2,10,4,12)(3,11,9,16)(5,13,8,14)(17,22)(18,27)(19,28)(20,31)(21,25)(23,24)(26,30)(29,32)(33,39,40,34)(35,44,42,37)(36,46,47,41)(38,43,48,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=g,d^2=h,e^2=ghi,f^2=h,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, 9113753);