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 | #40332 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(11,14)(16,18)(19,21)(23,24)(25,26,31,35)(27,36,29,32)(28,34,30,33)(37,39,38,40);(1,2,5,7)(3,4,8,6)(9,10,15,12)(11,16,14,18)(13,17,22,20)(19,23,21,24)(25,27,31,29)(26,32,35,36)(28,37,30,38)(33,39,34,40);(1,3,5,8)(2,6,7,4)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,28,31,30)(26,33,35,34)(27,37,29,38)(32,39,36,40);(1,3,5,8)(2,6,7,4)(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,26)(27,36)(28,33)(29,32)(30,34)(31,35)(37,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29,31,27)(26,32,35,36)(28,37,30,38)(33,40,34,39);(1,4)(2,3)(5,6)(7,8)(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,30,31,28)(26,34,35,33)(27,37,29,38)(32,39,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=hi,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275723);