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 | |
4161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,9,4,8)(6,10,14,13)(11,16,12,15)(17,18,23,27)(19,24,28,32)(20,21,25,30)(22,26,31,29)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19)(18,24)(20,22)(21,26)(23,28)(25,31)(27,32)(29,30)(34,40)(35,36)(42,45)(43,44);(6,14)(10,13)(11,12)(15,16)(19,28)(22,31)(24,32)(26,29)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15)(17,20)(18,21)(19,22)(23,25)(24,26)(27,30)(28,31)(29,32)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,5,7,2)(3,8,4,9)(6,13,14,10)(11,15,12,16)(17,21)(18,25)(19,29)(20,27)(22,24)(23,30)(26,28)(31,32)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,22)(18,26)(19,20)(21,24)(23,31)(25,28)(27,29)(30,32)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8256138);