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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,10)(9,16)(11,12)(17,18,22,23)(19,27,28,24)(20,25,29,32)(21,30,31,26)(33,34)(35,43)(36,41)(37,42)(38,39)(40,44)(45,48)(46,47);(1,2,6,7)(3,8,10,5)(4,9,14,16)(11,13,12,15)(18,23)(20,29)(24,27)(26,30)(35,44)(37,45)(40,43)(42,48);(17,19,22,28)(18,24,23,27)(20,26,29,30)(21,25,31,32)(33,35,39,44)(34,40,38,43)(36,45,47,37)(41,42,46,48);(1,3)(2,8)(4,12)(5,7)(6,10)(9,15)(11,14)(13,16)(17,19)(18,24)(20,30)(21,25)(22,28)(23,27)(26,29)(31,32)(33,36,39,47)(34,41,38,46)(35,37,44,45)(40,48,43,42);(1,4,6,14)(2,9,7,16)(3,11,10,12)(5,15,8,13)(17,20)(18,25)(19,26)(21,27)(22,29)(23,32)(24,31)(28,30)(33,37)(34,42)(35,36)(38,48)(39,45)(40,46)(41,43)(44,47);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,12,16,11)(17,21,22,31)(18,26,23,30)(19,25,28,32)(20,24,29,27)(33,38,39,34)(35,43,44,40)(36,46,47,41)(37,42,45,48)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=h,d^2=i,e^2=ghi,f^2=gi,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, 10124193);