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 | #57509 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(13,21)(16,22)(20,24)(25,26,30,31)(27,33,36,32)(28,29,37,38)(34,39,40,35);(1,2,5,3)(4,6,8,7)(12,19)(16,22)(18,23)(20,24)(26,31)(28,37)(32,33)(34,40);(2,3)(6,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26)(27,32)(28,29)(30,31)(33,36)(34,35)(37,38)(39,40);(2,3)(4,8)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,27,30,36)(26,32,31,33)(28,34,37,40)(29,35,38,39);(1,3,5,2)(4,7,8,6)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,28)(26,29)(27,34)(30,37)(31,38)(32,35)(33,39)(36,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29)(26,28)(27,35)(30,38)(31,37)(32,34)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012491);