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 | #45556 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4043 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)(19,27)(20,24)(21,25)(22,26)(23,30)(28,31)(29,32)(33,34,39,44)(35,45,47,40)(36,41,43,38)(37,42,48,46);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(17,19)(18,23)(20,28)(21,29)(22,30)(24,25)(26,27)(31,32)(33,34)(35,40)(36,41)(37,46)(38,43)(39,44)(42,48)(45,47);(2,4)(3,9)(10,12)(11,16)(17,20,22,29)(18,24,26,32)(19,28,30,21)(23,25,27,31)(33,35,39,47)(34,40,44,45)(36,48,43,37)(38,46,41,42);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,20)(18,24)(19,28)(21,30)(22,29)(23,25)(26,32)(27,31)(33,36,39,43)(34,41,44,38)(35,37,47,48)(40,46,45,42);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,21,22,28)(18,25,26,31)(19,29,30,20)(23,24,27,32)(33,37)(34,42)(35,36)(38,40)(39,48)(41,45)(43,47)(44,46);(1,6,7,15)(2,10,4,12)(3,11,9,16)(5,13,8,14)(17,18)(19,23)(20,24)(21,31)(22,26)(25,28)(27,30)(29,32)(33,38,39,41)(34,43,44,36)(35,46,47,42)(37,45,48,40)Polycyclic presentation:
2^9|a^2=hi,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, 9113752);