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 | |
4182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,4,8,7)(5,9,13,16)(11,12,15,14)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,3,6,8)(2,7,10,4)(5,11,13,15)(9,14,16,12)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(35,39)(36,42)(43,45)(44,47);(5,13)(9,16)(11,15)(12,14)(19,28)(22,31)(24,32)(26,29)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,15)(14,16)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4)(2,8)(3,10)(5,12)(6,7)(9,15)(11,16)(13,14)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,5,6,13)(2,9,10,16)(3,11,8,15)(4,12,7,14)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8313870);